Investigation on the effect of geometrical and geotechnical parameters on elongated offshore piles using fuzzy inference systems

Among numerous offshore structures used in oil extraction, jacket platforms are still the most favorable ones in shallow waters. In such structures, log piles are used to pin the substructure of the platform to the seabed. The pile’s geometrical and geotechnical properties are considered as the main parameters in designing these structures. In this study, ANSYS was used as the FE modeling software to study the geometrical and geotechnical properties of the offshore piles and their effects on supporting jacket platforms. For this purpose, the FE analysis has been done to provide the preliminary data for the fuzzy-logic post-process. The resulting data were implemented to create Fuzzy Inference System (FIS) classifications. The resultant data of the sensitivity analysis suggested that the orientation degree is the main factor in the pile’s geometrical behavior because piles which had the optimal operational degree of about 5° are more sustained. Finally, the results showed that the related fuzzified data supported the FE model and provided an insight for extended offshore pile designs.


Introduction
Oil and gas are the foundations of any developing nation. In order to extract these rich hydrocarbons from underground offshore deposits, many methods have been proposed and used. Jacket platforms are commonly used for such extractions. Fixed steel offshore platforms are generally composed of a deck in one or more levels, resting on the top of a steel jacket (Ferrante et al., 1980). The major structural details of the jacket skirt pile sleeve incorporated in the model include the jacket bracing, jacket leg, yoke plate, shear plate, and pile sleeves (Bao and Feng, 2011). These Jacket supporting piles have inclinations in them when being placed in the seabed. This dilation angle along with the pile's diameter and thickness is the most significant parameter affecting its geo-mechanical influence on the pile's bearing capacity (Fig. 1). In this study, first, a finite element software (ANSYS) was used to model the phenomena under various conditions, and then the extracted data from the software were fed to a fuzzy logic modelling software for predictions about uncertain system. In the past few years, the fuzzy inference system (FIS) (expressed in terms of fuzzy rules) has been employed in the prediction of uncertain systems because its application does not require knowledge of the underlying physical process as a precondition (Bateni and Jeng, 2007). The utilization of fuzzy logic system is a main classification tool to present predictions for the pile's unmeasured behaviors. Fuzzy logic system (FLS) and probabilistic analysis were developed for linguistic classification of a system and interactions between its variables (Mojtahedi et al., 2011). Fuzzy set theory is a nonlinear universal mapping with consideration of the uncertainties associated with ambiguity and imprecision due to the lack of information about the system (Mojtahedi et al., 2012).
As mentioned before, jackets play a vital role in the offshore industry in field development and operation, with proven flexibility and cost effectiveness (Korzani and Aghakouchak, 2015). These structures are attached to the sea floor by long piles. In order to define the bearing capacity of the piles, their angle of inclination to the horizon, their diameter, and length are generally examined. Insufficient design of the foundations of such structures can result in disasters. For instance, in 2008, an offshore platform in an area in the East Cameron was hit by hurricane "Ike" and failed due to the axial failure of pile, which caused the platform leaning. Coincidentally, on Sep. 7, 2010, the No. 3 drilling platform of Shengli Oilfield on the East China Sea was tilted by strong winds during typhoon "Malou" (Chen et al., 2015).
For this study, through the finite element analysis, piles were positioned in various soil environments and the data collected have been studied to introduce the optimum geometric solution to the supporting pile. In general, nonlinear pile-soil interaction is the most important source of nonlinear response of offshore platforms due to designing environmental loads. Hence, the lateral cyclic deflection of the platform using cyclic backbone curves is considerably higher than the corresponding results under monotonic loads, and the pile responses (deflections, shear forces and bending moments) by using cyclic curves are more sensitive to cyclic loads than those by using static backbone curves (Memarpour et al., 2012).
Stress distribution within such a large platform structure is a dominant factor in the design procedure of an offshore pile. The loading of an offshore structure consists of two vertical structural loads and lateral wave loads. The peak displacement of the pile significantly decreases as the wave period increases, In other words, shorter wave periods have a more critical effect on the pile displacement. (Eicher et al., 2003).
In general, the loading of an offshore structure consists of two components, the vertical structural loads and the lateral wave loads. The combination of these two loading components has a significant impact on how the pile reacts and the way that the stress levels are distributed throughout the pile.
The main aim of this study is to investigate the effects of the combined loads on an offshore pile with respect to the various loading parameters. Results show that the orientation degree must be considered as the main parameter to evaluate the pile's geometrical behavior. Nevertheless, as noticed in the literature, many practical problems and difficulties caused by various types of uncertainties make it a complicated issue. On the other hand, there is no report that directly focuses on the effects of the uncertainties. The main scope of this study is to circumvent these major problems by utilizing the concept of well-known techniques for offshore platform pile design. Consequently, this problem is the main objective for this study to scrutinize and develop a robust system that can consider the uncertainties and the effects of nonlinearity on the relations between the mentioned parameters.
For this purpose, a combined algorithmic incorporation of fuzzy logic system (FL or FIS) and finite element modeling (FE modeling) was used to study the effects of the geometrical and geotechnical parameters of offshore piles on the supporting jacket platforms. This method has been proposed to consider the uncertainty problems with the consideration of benefits and some shortcomings in the literature. The concept of the method was developed based on the nonlinear geometric characteristics of the related system responses and the nonlinearities in the nature of the evaluated system. The observed results imply that the method can be served as an efficient technique to detect the effects of the sensitive parameters.

Finite element modeling
To create a finite element model in the ANSYS software, the first step is to construct a media for the volumetric geometry of the models and furthermore for the elements (see Fig. 2). Solid 186 has been chosen to place the model mainly because solid 186 has 20 nodes in all three dimensions and is capable of studying element behaviors with multiple degrees of freedom.

Geometrical FEM modeling
Every element regarding its triplet degrees of freedom (its movement in X, Y and Z directions) is introduced to the system with the maximum number of 20 nodes (Fig. 3). In order to construct the model, a parameter designated as "teta" (in degrees) which is the orientation angle of the pile to the XY plate was defined. Next, the length of the pile is  (Whitehouse, 1998). Ali AMINFAR et al. China Ocean Eng., 2017, Vol. 31, No. 3, P. 378-388 379 introduced to the software (30 m for this scenario). The length of the soil media is then introduced in the Y direction.
In order to define the width of the soil media, the orientation of the pile was considered. It means that, as the pile orientates in the X direction the geometrical volume needed to contain the pile will increase. The elevation is in the Z direction and has been set to its maximum in all scenarios by taking the length of the pile and adding an extra 4 m to assure that the behavior of the model will not exceed by any increase resulting from elevation changes. Now a cube can be introduced to the software to define the soil volume with the given dimensions in the software. A cylindrical coordinate system with its origin in the pile's end with the coordinates of (4, 2, 2) (the location of the pile at the bottom of the soil) and its axis having teta degrees orientation with the Z axis of the original coordinate system is introduced to the software to create the pile. At the origin of the newly created coordinate system dual circles with the diameters of 0.23 and 0.25 m for the first type, 0.28 and 0.30 m for the second and 0.33 and 0.35 m for the last type of piles are drawn. The first and second diameters for all of the three types are the interior and exterior diameters of a cylinder that is yet to be modeled. The difference of each set of the dual circles represents the thickness of the piles. The circles are then extruded to 30 m to create two cylinders. The cylinders are subtracted from each other to create a thin shell volume described in this article as the pile. After the subtraction, two volumes are created, one is the volume defined as the pile and the second volume is the inner part of the pile that in this work will be filled with soil ( Fig. 4).

Mesh type acquisition
The next step after defining the volumes is the appliance of the meshing. Tetrahedral meshes are chosen for this study due to the model's non-uniform geometry and its compliance with the Solid 186 environmental assumptions. The pile and the soil media have been meshed individually and then element contact methods have been applied to attach one to another.

Material properties
The pile has been constructed in accordance with the ASTM A36 steel standard with a specific weight of 7700 kg/m 3 (ASTM, 2014). Additional parameters regarding the ASTM-A36 steel are illustrated in Table 1.
In order to define the soil that we put the pile in, we need to choose a behavioral modal for the soil. It is needless to say that soil has nonlinear behaviors when it comes to modelling. As a result, many behavioral modals have been developed for this phenomenon. Among the many models, the Drucker-Prager/Cap is utilized in this study.
The Drucker-Prager/Cap plasticity model has been widely used in the finite element analysis programs for a variety of geotechnical engineering applications (Helwany, 2007). The cap model is more than appropriate to soil behavior because it is capable of considering the effect of stress history, stress path, dilatancy, and the effect of the intermediate principal stress. The yield surface of the modified Drucker-Prager/Cap plasticity model consists of three parts: a Drucker-Prager shear failure surface, an elliptical cap (which intersects the mean effective stress axis at a right angle) and a smooth transition region between the shear failure surface and the cap (Fig. 5).
The soil's elastic behavior is modeled as linear elastic using the generalized Hooke's law. In accordance with the Hooke's law, which the bulk elastic stiffness increases as the material undertakes compression, an elasticity model was used to calculate the elastic strains described in Eq. (1). The onset of plastic behavior is determined by the Drucker-Prager failure (slip) surface and the cap yield surface. The Drucker-Prager failure surface is expressed as: where, β is the soil's angle of friction and d is its cohesion in the p-t plane (p and t represent the normal and shear stress, respectively), as indicated in Fig. 5. The value of K is used to determine the projection of the yield surface on the Π-plane (as seen in Figs. 5 and 6). k is a constant that shapes and ensures the convexity of the yield surface and has a range from 0.778 to 1. Setting k to 1 causes the yield surface to be independent of the third stress invariant. e 0 and p′ are the initial consolidation curve ratio and the mean effective stress. As shown in the figure, the cap yield surface is an ellipse with the eccentricity R in the p-t plane. The cap yield surface is dependent on the third stress invariant, r, in the deviatoric plane as shown in Eqs. (3) and (4) (Fig. 5) (Helwany, 2007).
where p 1 , p 2 and p 3 are the stress tensors. So, J 1 and J 2 are the first and second invariants of the stress tensor. As a result, J 2D and J 3D are the second and third deviatoric stress tensors. The cap surface expands or shrinks as a function of the volumetric plastic strain. When the stress state causes the yielding on the cap, the volumetric plastic strain compaction results, causing the cap to expand.
However, when the stress state results in the yielding on the Drucker-Prager shear failure surface, the volumetric plastic dilation occurs that causes the cap to shrink. The cap yield surface is as: The parameter R is a material parameter that controls the shape of the cap and α is a small number (usually between 0.01 and 0.05) used to define a smooth transition surface between the Drucker-Prager shear failure surface and the cap: where p a is an evolution parameter that controls the expanding-shrinking behavior as a function of the volumetric plastic strain. The expanding-shrinking behavior is no more than a piecewise linear function relating the mean effective (yield) stress p b and the volumetric plastic strain p b =(ε vol pl ) (Fig. 6). This function is obtainable from the results of an isotropic consolidation test with multiple unloading-reloading cycles. As a result, the evolution parameter, p a , is as (Helwany, 2007): For the Drucker-Prager failure (slip) surface and the transition yield surface, a nonassociated flow that is identical to the yield surface (i.e., an associated flow) is assumed.
The shape of the flow potential in the p-t plane is different from the yield surface as shown in Fig. 7. In the cap region, the elliptical flow potential surface is expressed as:   Ali AMINFAR et al. China Ocean Eng., 2017, Vol. 31, No. 3, P. 378-388 381 The elliptical flow potential surface portion in the Drucker-Prager failure and transition regions is defined as: The two elliptical sections of the surface, G c and G s , provide a continuous potential surface. The material stiffness matrix is not symmetric due to the use of a nonassociated flow in the model. However, an asymmetrical solver is needed in the association with the cap model (Fig. 8).
In order to determine the parameters d and β, the results of the minimum of three triaxial compression tests are required. The at-failure conditions taking the results of the tests can be plotted in the p-t plane. A straight line is then best fitted to the three (or more) data points. The intersection of the line with the t-axis is d and the slope of the line is β. To evaluate the hardening-softening law as a piecewise linear function relating the hydrostatic compression yield stress p b and the corresponding volumetric plastic strain p b =(ε vol pl ) as mentioned there has to be data from at least one isotropic consolidation test with several unloading-reloading cycles (Fig. 5). The unloading-reloading slope is used to calculate the volumetric elastic strain that should be subtracted from the volumetric total strain to calculate the volumetric plastic strain (Helwany, 2007).
After the brief introductions to the Drucker-Prager/Cap model, the parameters needed to define the soils' nonlinear characteristics in accordance with the model are illustrated in Table 2. Three types of soils have been defined to the software, a very cohesive soil (φ & ψ=0, Type A), non-cohesive (C=0, Type C) and a mixture of both (Type B).

Applying loads
After applying the mesh, a surface load of 2E+08 (N/m 2 ) was implemented to the upper ring of the pile. Then, the solver was applied to the model.

FEM verification
Hence, the data gathered from the FEM software are mainly analytical; the models need to be verified with previous laboratory evaluations. Zou et al. have studied long pipe piles in multiple layered soil environments. They have conducted actual field and laboratory tests on such pile systems and have later mathematically modeled their system based on the element free Galerkin method (EFGM). Their pipe pile had the diameter of 1 m and the length of 60 m. Although their pile was made of reinforced concrete, they simplified their data for computational reasons as in using weight average values for the pile's elastic modules and tensile strength. In the present article, the material modals for the pile were adopted with respect to the work conducted by Zou et al. The soil layers and their geotechnical parameters were extracted from the proposed article (Zou et al., 2007;Zou and Zhao, 2013). The results from the finite element analysis conducted by the authors and the results illustrated by Zou et al. are depicted in Fig. 9.
The presented surface from the FEM modeling has smaller than the maximum of 8% difference from the measured values from Zou et al.'s actual model and it is slightly more accurate than Zou et al.'s EFGM results. Consequently, the data from the FEM model can be verified to be highly accurate and acceptable.

Introduction to the fuzzy inference systems
Fuzzy logic is a logical system, which is an extension of multivalued logic. The purpose of the fuzzy logic is to map an input space onto an output space, and the primary mechanism for doing this is a list of if-then statements called rules. All rules are evaluated in parallel, and the order of the rules is unimportant. In other words, the fuzzy inference is a method that interprets the values in the input vector and is based on some set of rules, which assigns values to the output vector. As a part of Matlab, the fuzzy logic toolbox software can perform as a tool for solving problems with the fuzzy logic (Zandi Goharrizi et al., 2014). In general, any fuzzy system contains the following steps: (1) Inference system definition based on data: Analysis operation is applied by the fuzzy inference engine. There are several fuzzy inference systems which can be utilized for this purpose, such as Sugeno and Mamdani fuzzy inference systems which are two of the most important ones; (2) Membership function definition: Input information is made as fuzzy data by membership functions. This step is known as fuzzification; (3) Inference rule definition and combination; (4) Obtaining results and defuzzification if being needed (Zandi Goharrizi et al., 2014).

Creating the fuzzy logic system
In the following article, Mamdani fuzzy inference systems for investigation have been utilized. Fig. 10 illustrates the Mamdani fuzzy inference systems for the settlement analysis. For the other four parameters' extracted from the FEM software "ANSYS", similar systems have been utilized.
The algorithm designed for this system consists of three fuzzy input variables. These inputs are the orientation degrees, pile diameters, and soil types that are the same parameters' from the FEM analysis input (Fig. 11). In order to introduce the soil type in a quantitative way to the FIS software, each soil type was valued by a block with the full membership functionality and the width of 0.3. The blocks are typical for all soil types and only differ in their initial and final placements. Therefore, as seen in Fig. 10, the first part from 0 to 0.3 represents the non-cohesive soil type. The last part from 0.6 to 0.9 is the cohesive soil type and the part between them represents a mixture of both soil types. In order to establish the best type and boundary for the surfaces of the input variables, trial and error procedures were conducted.
Output membership functions for one of the data sets such as in the variable "Settlement" are presented in Fig. 12.
After going through a process of trial and error to select the type of the membership functions (if then assumptions), the rules defining the behavior of the system have now been developed (Fig. 13). Finally, defuzzification of the obtained output and the results of the purpose have been started as stated.

FIS verification
Hence, to verify the operational eligibility of the FIS to model, Zou et al.'s actual laboratory data are modeled using the software and the results are illustrated in Fig. 9. The measured curve is clustered and then the fuzzyfication process has been applied to the whole curve with regard to each cluster. The given outcome suggests that the FIS solution has the maximum of 5% deflection to Zou et al.'s calculations. As a result, this method is adequately accurate to perform the given task (Zou et al., 2007;Zou and Zhao, 2013).  Ali AMINFAR et al. China Ocean Eng., 2017, Vol. 31, No. 3, P. 378-388 383 3 Results and discussion

FEM results
The maximum von Misses stress increases with the increase of piles diameter. This growth is extremely rapid for piles facing orientations more than 9° and for smaller orientations is less tangible. By increasing the orientation of the pile, the maximum von Misses stress grows rapidly to 6 times its initial loading mainly due to the deformation of the pile in certain areas. In the inclination degrees more than 18°, the change in the pile diameter has more effect. Nevertheless, the increasing trend in the stress within cohesive soils is slightly lower for soils with grained particulates (Fig. 14).
The maximum von Misses elastic strain is higher in the piles with bigger diameters. Piles with the orientation smal-ler than 5°, exhibit small difference when the pile diameter changes. This increasing trend is more noticeable in the orientations that are larger than 7°. However, this increasing trend seems to be the same for all pile diameters. The maximum elastic von Misses strain gradually increases to its double amount at an orientation degree of about 7°. Nevertheless, this growth of the piles maximum strain reaches its peak at 8 times its initial value. Cohesive soils exhibit lower strain dilations (Fig. 15).
The increase rate of the maximum displacement (total displacement) is less sensitive to the orientation smaller than 6° by exhibiting less fluctuation in values and has its minimum of 4° in the orientation. By contrast, for the orientation larger than 7°, this phenomenon has outcomes that are severe by showing the increase on larger diameters. In the inclination more than 7°, the increase in the maximum displacement vector sum is more severe growing to 3 times the minimum displacement. The maximum displacement vector sum surfaces suggest slightly more displacements for cohesive soils; however, at higher orientations, the displacement margin grows rapidly (Fig. 16).
The piles lateral response suggests a parabolic shape with the minimum of 4° in the orientation and has less sens-     itivity to the diameter changes; which means that by the increase of the diameter, the pile has shown slight fluctuations. This pattern is vital on the orientation more than 7°, because in such scenarios the difference between the responses of the pile and the loading is notably considerable with respect to the diameter changes. All soil types behave similarly at low orientations. However, when the orientation is larger than 9°, cohesive soils have lower lateral dilations (Fig. 17).
The maximum settlement value decreases as the weight of the pile itself decreases. In other words, the pile with small diameter has slightly better behavior when it reaches the maximum displacements in the vertical direction. The settlements' surface consists of two phases, the first one with the orientation degrees from 1° to 9° and the second from angles more than 9°. The first phase has a constant trend, which means that the settlements in this phase do not face severe fluctuations. However, on the orientation angles from 4° to 5°, the settlements are at their minimum. In the second phase, a sharp increase is notable for the settlement pattern. The increasing pattern of the settlement itself has a linear trend, and changes in the soil types are very significant at low orientation degrees with more settlements for cohesive soils. (Fig. 18).
In Figs. 19 and 20, the cross sections have been cut within the deformed shape of the pile in the soil volume. These cross sections not only illustrate the actual deformation shape of the pile, but also they demonstrate that the soil inside the pile has no or little impact on the outcome of the pile's response for all scenarios.

FIS results
Two types of results can be extracted from the fuzzy-logic toolbox. First, they are the visual surfaces as illustrated in Fig. 21.    Ali AMINFAR et al. China Ocean Eng., 2017, Vol. 31, No. 3, P. 378-388 These surfaces can be drawn to compare all of the systems defining parameters 3 at a time and because they are visual, they can produce qualitative comparisons for the models behaviors. The results of next type are quantitative ones that give actual numerical data and are acquirable from the rule viewer toolbox of the software (Fig. 22).
In order to authenticate the results from FIS, first they must be verified. A set of data from the FEM analysis were chosen, and then they were compared with the results from the fuzzy logic system. The crosscheck between the data is presented in Table 3.
In order to examine the prediction accuracy of the forecasted data from the FIS model, the mean absolute percentage errors (MAPE) were calculated for all of the 5 data sets. The MAPE can be calculated through Eq. (10).
where N is the total number of the fitted points, A n is the actual forecasts and P n is the predictions. The results are illustrated in Table 4. All of the predicted parameters except the maximum von Misses stress have the MAPE smaller than 10%. For the maximum von Misses stress, the MAPE is 17% due to the nature of the stress values that vary in a large period of 3.34E+09 (N/m 2 ) from the maximum to minimum values. As a result, the predictions are in an acceptable accordance with the actual FEM results and they can be used to predict pile's behaviors.

Conclusions
The finite element modelling (FEM) and fuzzy inference system (FIS) results are combined to present the following findings: (1) The pile's orientation degree is the primary parameter to evaluate the pile's geometrical behavior. Such piles behave optimally at the operational orientation of about 5°. Any variations in the optimum orientation degree may result in the increase of the pile's instability without respect to the geotechnical properties of the soil surrounding the pile.
(2) Cohesive soils behave poor when compared with the grained soils no matter what the piles geometric conditions are.
(4) Piles with the orientation more than 10° have to be restricted to certain structures (i.e. dolphins, waterfronts and etc.). In the design and installation phases of such structures, extreme caution must be exercised.
(5) The concept of the FIS was used to study the effects of various geotechnical and geometrical parameters on the piles' supporting jacket platforms. Based on the numerical results, it was observed that the FIS could be used as a valuable tool to predict the effects of such parameters with an acceptable error margin on the FEM solutions.