Preliminary Dynamic and Fatigue Analysis Using Directional Spectra
Authors: Marshall P.W. 
Published: December 2007

The historical development of spectral analysis is reviewed and a concise linear theory is derived for spectral wave forces and resulting hot-spot stresses, including directionality, and spreading. The theory is applied to preliminary dynamic analysis and spectral fatigue damage calculations for a proposed 1,000-ft platform in the Gulf of Mexico.

Introduction
The traditional design approach for fixed offshore structures involves the use of a higher-order, regular-wave theory such as Stokes' fifth. When the design is governed by applied force from the single largest wave that completely envelops the structure and nonlinear drag or shallow-water effects are important, this approach is appropriate, with reasonable ease of computation and order-of-magnitude accuracy.
However, for deep-water structures for which dynamic response and fatigue are important, this approach no longer suffices. It is particularly unrealistic to consider that all the wave energy is concentrated at one frequency. Beginning in the early 1950's, St. Denis and Pierson pioneered the application of spectral-energy theory to the pioneered the application of spectral-energy theory to the analysis of ship motions.
Using leads developed by Borgman and Nath and Harleman, similar spectral techniques have been applied since 1967 in the design of Shell Development Co.'s caissons, deep-water templates, and in rough-sea designs, An early fatigue analysis can be found in Ref.4. A more rigorous formulation by Penzein et al. has been implemented recently as Program TOWER. In both these approaches, drag forces are retained in a variable linearization along with hydrodynamic damping; however, this is done at the expense of a grossly simplified representation of the structure. Calibration is based on spectral analysis of forces on instrumented cylinders and on comparisons of predicted extremes with the traditional deterministic approach. Fatigue criteria for use in random loading in the offshore environment have been developed (see Figs. 1A and 1B), and results have been calibrated by hindcasting existing structures.
However, this was not to be the final answer. Reviews of recent in-house fatigue analyses for fixed and floating structures in North Sea service indicated that using directional spectra might yield a substantial reduction in computed forces. One structure, a twin-torpedo semisubmersible, was particularly sensitive to perfectly broadside prying waves, so that fatigue-life calculations based on prying waves, so that fatigue-life calculations based on unidirectional waves might be considered unreasonably pessimistic. pessimistic. It has long been recognized that real waves are not infinitely long-crested and that directional spectra, which account for the spreading of wave energy by direction as well as by frequency, are required for a complete statistical distribution of the real sea. In the late 1960's, work performed at Shell's Bellaire Research Center suggested performed at Shell's Bellaire Research Center suggested that considering directional spectra could qualitatively explain a number of anomalies in existing wave-force measurements and analyses, including the C, shotgun plot that typically results from deterministic attempts at plot that typically results from deterministic attempts at wave-force calibration.
Borgman investigated the complexities of deriving directional spectra from arrays of wave staffs, and is presently working on methods for reducing combined presently working on methods for reducing combined wave-staff/current-meter data to directional spectra. Concurrently, the analysis phase of the offshore industry's cooperative Ocean Data Gathering Program (ODGP) has involved directional-spectra hindcasts by Cardone et al.
JPT
P. 715