Closed-Form Expressions for Determining the Fatigue Damage of Structures Due to Ocean Waves
Authors: Nolte K.G. , Hansford J.E. 
Published: December 2007

Abstract
Closed-form mathematical expressions are derived for the fatigue damage of structures because of ocean waves. The expressions incorporate relationships between wave height and stress range, between stress range and number of cycles to failure (i.e., a fatigue curve), and the probability distribution for the occurrence of wave heights. The expressions can be used to predict the fatigue damage resulting from a single sea state, from a storm, or during the service life of a structure. Also, the fatigue life of a structural element can be determined directly from the stress range resulting from the design wave, or conversely, an "allowable stress range" can be determined for the design wave that will insure a specified fatigue life. Example applications are given for areas having wave climates similar to the North Sea and the Gulf of Mexico.

Introduction
Ocean waves encountering a structure cause the stress in each structural element to cycle. The accumulated effect of the stress cycles can cause element of an improperly designed structure to fail because of fatigue. In general, from the standpoint of fatigue damage, there are two categories of structures:structures with a response to ocean waves that are significantly affected by dynamic amplification andstructures without significant dynamic amplification. This paper will be restricted to the last category of structures for which dynamic amplification can be neglected for fatigue analyses.
The current state-of-practice for the fatigue analysis of ocean structures is summarized by Marshall. The analysis of the fatigue life of a structure requires a lengthy numerical calculation procedure which utilizes: procedure which utilizes:The appropriate curve for the structural element being considered. The curve relates the number of cycles to failure (N) for a cyclic stress range (), having a constant amplitude (see Fig. 1).The Palmgren-Miner rule that predicts the cumulative effect of stress cycles having different amplitudes.An empirical relationship between the wave height encountering the structure and the stress range induced in the structural element for all wave heights that the structure will encounter.A statistical description of the occurrence rates for wave heights that the structure will encounter during its life.
The numerical analysis consists of breaking the range of wave heights into discrete bands and determining the number of waves that will occur in each height band. For each height band, the stress range corresponding to the mid-band height is determined for the structural element. Then, the curve is used to determine the fatigue damage for the element because of the stress range associated with each height band. The final step is to accumulate the damage for all height bands using the Palmgren-Miner rule. This procedure can be undertaken manually in a straightforward, time-consuming manner or can be programmed for a digital computer.
As a result of the time required to undertake the procedure for each fatigue-prone element, a fatigue procedure for each fatigue-prone element, a fatigue analysis rarely is undertaken during the initial design phase of a structure. Instead, during the initial design, members are sized only on the basis of ultimate loading and a fatigue analysis is performed after the initial design. If the analysis performed after the initial design. If the analysis indicates potential fatigue problems, the design procedure is recycled. procedure is recycled. In this paper, closed-form expressions are derived that permit a fatigue analysis to be performed quickly, compared with the equivalent performed quickly, compared with the equivalent numerical procedure outlined above. The expressions can be applied manually or as part of a stress analysis computer program. As a result, fatigue considerations can be integrated into the initial phase of the design along with ultimate stress phase of the design along with ultimate stress considerations. The closed-form equations yield results that are no better nor worse, but equal to a careful numerical analysis. The sufficiency of such a procedure depends on the situation and must be decided on a case-to-case basis.
SPEJ
P. 431