Dynamic Response of an Offshore Pile, a Poro-Elastic Seabed and Seawater due to Water Waves

In this study, a coupled model is developed to address the dynamic response of an offshore pile to linear water waves. In the coupled model, the pile and the seabed are treated as a saturated poro-elastic medium described by Biot’s dynamic theory, while the seawater is considered as a conventional acoustic medium. To establish the coupled model, boundary element formulations for saturated porous media are derived for the pile and the seabed, respectively, and the acoustic boundary element formulation is constructed for the seawater. The coupled boundary element model is obtained using these three boundary element formulations as well as the continuity conditions along the interfaces between the pile, the seabed and the seawater. In the model, linear water waves are considered as an external load and its load is evaluated via the wave function expansion method. Numerical results shows that increment of the modulus ratio between the pile and the seabed can reduce the horizontal displacement of the pile and the pore pressure of the seabed around the pile. Also, the maximum pore pressure of the seabed usually occurs at the upper part of the seabed around the pile. INTRODUCTION It is well-known that sea waves can cause enormous loads on the offshore structures, particularly, in rough weather conditions. Thus, motivated by the need of safe recovery of oil and natural gas in the offshore region, ocean engineers have paid much attenuation to the determination of the wave loads on offshore structures. However, due to incomplete knowledge concerning offshore structures as well as their environmental factors and resulting inappropriate designs, some marine structure failure incidents have occurred recent years [1-3]. Therefore, a safe design of an offshore structure remains a challenge for ocean engineers. The pile foundation is a very common and important structure for offshore engineering, which can be used as the foundation for offshore wind farms, offshore platforms, offshore ports, long-span cross-bay bridges etc. To date, numerous researches concerning offshore piles and water waves have been carried out. For example, the effects of various waves on offshore piles have been investigated by many investigations [4-6]. The offshore pile capacity and stresses and strain of a pile due to wave loads were addressed in [7, 8], respectively. Mitwally and Novak [9] used a linear analysis to address dynamic interaction between pile–soil–pile, when the system is subjected to a random wave loading. Moreover, using the concept of dynamic p–y curves, the response of fixed offshore platforms to wave and current loading when taking into account the soil–pile interaction was investigated by Mostafa & Naggar [10]. It is worthwhile stressing that existing researches about offshore piles, the seabed and the seawater mainly focused *Address correspondence to this author at the Department of Civil Engineering, JiangSu University, Zhenjiang, Jiangsu, 212013, P.R. China Email: [email protected] and Division of Civil Engineering, School of Engineering, Physics and Mathematics, University of Dundee, Dundee, DD1 4HN, Scotland, UK; Email: [email protected] on separate aspects of the problem, rather than treated them as a coupled system. However, in natural ocean environments, the pile, the seabed and the seawater are coupled. Therefore, response of the pile is related to those of the seabed and the seawater. Furthermore, besides the design for the pile itself, a complete design for offshore pile foundation should also include the design for the seabed, which aims to guarantee the seabed have enough strength to resist liquefaction and shear failures. Obviously, the design of the seabed will unavoidably involve detailed knowledge concerning the stresses, the deformation and the pore pressure of the seabed, which can only be available via a coupled analysis of the pile-seabed-seawater system. Consequently, a coupled model for the offshore pile, the seabed and the seawater is crucial for a successful design for the offshore pile foundation and its surrounding seabed. Another limitation of existing researches concerning the pile-seabed interaction is that only single phase medium model is used to describe the seabed. Nevertheless, it is well-known that the seabed is a porous medium saturated by seawater and more importantly, pore fluid of the seabed plays a very important role in the liquefaction and the shear failure of the seabed. Thus, an appropriate coupled model for the pile-seabed-seawater system should be able to account for the role of the pore fluid of the seabed. In this study, a coupled model is developed to investigate the interaction between the pile, the seabed and the seawater, when the system is subjected to linear water waves. The main contributions of this paper are as follows. First, we propose the idea to treat the pile, the seabed, the seawater as a whole system, which has not appeared in the literature to date. Secondly, the coupled model for the pile, the poro-elastic seabed and the seawater is established technically for the first time by combining the BEM formulations for the pile, the seabed and seawater via the continuity conditions along their interfaces. Also, the treatment of the Cauchy type integral for the half space seawater belongs to our own contribution. 100 The Open Civil Engineering Journal, 2008, Volume 2 Lu and Jeng The remainder of this paper is organized as follows. First, Airy linear wave theory [11] is used to calculate the external load applied to the system due to linear water waves. Second, Biot’s theory [12, 13] is outlined. Then, the boundary element formulations for the poroelastic medium and the acoustic medium are established. Three BEM formulations for the pile, the seabed and the seawater are coupled together to form the coupled BEM model for the system. Numerical examples are presented to demonstrate the capacity of the proposed model. EXTERNAL LOADS APPLIED TO THE SYSTEM DUE TO LINEAR WATER WAVES In this study, linear water waves are considered as the external load applied to the pile and the seabed. When evaluating the external load due to linear water waves, the pile and the seabed are assumed to be fixed. The external load obtained in this section will be applied to the pile-seabedseawater system to calculate the response of the system via the coupled model. As only frequency domain response of the pile-seabedseawater system is considered in this paper, as a result, only the monochromatic linear water wave or the frequency domain linear water wave is involved here. The waves which have contribution to the external load applied to the pile and the seabed consist of two parts: the incident wave and the scattered wave. Thus, the velocity potential for the total external load has the following expression, ( ) ( ) ˆ ˆ ˆ I S = + (1) where ( ) ˆ I and ( ) ˆ S are the velocity potentials for the incident wave and the scattered wave, respectively and a caret above a variable denotes the frequency domain. The velocity potential for the incident plane wave is given by ( ) ( ) 2 ˆ [ ( )] ( , ) I i t ky iky A ch k z h e G z t e = = . (2) where 2 h is the depth of the seawater, and A is determined by the expression 2 2 cosh gH A i kh = (3) where H is the wave height of the incident wave and g is the gravitational acceleration. The angular frequency and the wave-number of the incident wave fulfill the following dispersion relation 2 2 ( ) gk th kh = . (4) It is noted that downward direction of z-axis is assumed to be positive, as depicted in Fig. (1). To determine the scattered wave field analytically, it is appropriate to study the current problem in the cylindrical coordinate system as shown in Fig. (1). Thus, the incident wave given by (2) can be expanded into the following series expression [11]