# Nonlinear and Stochastic Dynamics of Compliant  Offshore Structures

The purpose of this monograph is to show how a compliant offshore structure in an ocean environment can be modeled in two and three dimensions. The monograph is divided into five parts. Chapter 1 provides the engineering motivation for this work, that is, offshore structures. These are very complex structures used for a variety of applications. It is possible to use beam models to initially study their dynamics. Chapter 2 is a review of variational methods, and thus includes the topics: principle of virtual work, D'Alembert's principle, Lagrange's equation, Hamilton's principle, and the extended Hamilton's principle. These methods are used to derive the equations of motion throughout this monograph.
Chapter 3 is a review of existing transverse beam models. They are the Euler-Bernoulli, Rayleigh, shear and Timoshenko models.
The equations of motion are derived and solved analytically using the extended Hamilton's principle, as outlined in Chapter 2. For engineering purposes, the natural frequencies of the beam models are presented graphically as functions of normalized wave number and geometrical and physical parameters. Beam models are useful as representations of complex structures. In Chapter 4, a fluid force that is representative of those that act on offshore structures is formulated. The environmental load due to ocean current and random waves is obtained using Morison's equation. The random waves are formulated using the Pierson-Moskowitz spectrum with the Airy linear wave theory.

In Chapter 5, the possibility of beam axial displacement is added to the formulation. The equations of motion are now nonlinear and coupled. The free, damped-free, harmonically forced, and randomly forced responses are considered in this chapter. This model can predict the axial displacement, longitudinal extension, and normal stress in addition to the transverse displacement. However, this is a two-dimensional model, which can only accommodate two-dimensional forcing. The fluid forces in an ocean environment are inherently three-dimensional. Therefore, a three-dimensional beam model is considered in Chapter 6. Both rigid and elastic models are considered. The elastic model is derived assuming moderate rotation. Free and forced responses are studied. A harmonic force and a constant force are applied in perpendicular directions to simulate possible vortex-induced fluid forcing resulting from the interaction of a constant current with the beam-like structure. Chapter 7 provides a summary and conclusions. Several appendices provide some additional details.

The reader may ask how representative beam models are of the massive offshore structures one finds in the ocean, in applications such as oil drilling, docking facilities and communications junctions. The design of any such complex structure must begin with an understanding of the physical processes of the design environment and how these interact with the potential structure. This understanding is best obtained using reduced-order models that are faithful to the most important underlying physics. Beam models of the type formulated here are ideally suited for the studies at hand. Their analysis signals the analyst and engineer where key issues must be further analyzed in more detail with higher-order models, eventually leading to the full scale model and design. Therefore, the analytical and computational process described in this monograph should be viewed as a first step in a very intricate and complex process. These are valuable exercises.