376 J.M. Rinker and H.P. Gavin but rather modeled the SN uncertainty directly in the limit state formulation. For example, Wirsching uses the Palmgren– Miner rule and an SN power-law relationship to formulation a stochastic damage function [7]. Zhao et al. follow a similar formulation to evaluate the fatigue reliability of steel bridge components [8]. Veers also followed the same formulation as Wirsching, generating a stochastic damage function for the lifetime of a wind turbine blade [9]. In this paper, we compare the PSF methodology that is currently recommended in the IEC design standard with a formulation of the damage index that directly models fatigue resistance uncertainties. Our proposed formulation follows a procedure that is similar to those laid out above, but here the damage indices are manipulated such that there is a direct relationship between the partial safety factors in the IEC standard and the stochastic parameter in the power-law relationship. This relationship is beneficial because it allows a direct comparison between the IEC standard and our proposed method, which could lead to an improved method for calibrating the safety factors. 37.2 Theory Both the IEC formulation and our method use the Palmgren–Miner damage accumulation rule to formulate an expression for the expected amount of damage that will accumulate over a turbine’s lifetime; this formulation is summarized in Sect. 37.2.1. The differences between the two methods arise when the expression for the cycles to failure is replaced with an SN-relationship, which is the focus of Sect. 37.2.2. 37.2.1 Formulation of the Damage Function The Palmgren–Miner linear damage accumulation rule quantifies the fatigue damage accumulation, D, in an object under random loading. It is a scalar index that grows with the number of cycles ni that occur at a stress level Si , mormalized by the number of cycles to fatigue N.Si / at constant stress amplitude: DDX i ni .Si / N.Si/ : (37.1) The load range values Si can be calculated from a data record using a cycle-counting method such as the Downing and Socie algorithm [10]. In this equation, Ddenotes the damage that is accumulated from the collection of load cycles Si , and failure occurs when D 1. In this equation, the load ranges Si can be determined from a particular data record, in which case Dis a deterministic value that represents the damage accumulated over the time of data acquisition. Alternatively, the load ranges could be modeled stochastically, in which case each load range would have a certain probability of occurring in a given period of time. In this case, the distribution of load ranges can be described by a cycle density functionnS.sjTS;U/, where U is the mean wind speed duringTS. This density function is similar to a probability density function (PDF), but it is scaled such that the total area beneath the curve is equal to the expected number of cycles that will occur in a record of length TS. Note that an uppercase U or S indicates a stochastic variable, whereas a lowercase u or s indicates a deterministic value of that variable. Converting Eq. (37.1) into continuous form yields the damage accumulated over time TS, as a function of the mean wind speed: D.U/ DZ 1 0 nS.sjTS;U/ N.s/ ds: (37.2) For simplicity, both the IEC standard and the proposed formulation eliminate the wind stochasticity by taking the expected value with respect to the mean wind speed. Additionally, because damage accumulation is linear, the expected amount of damage that will be accumulated during a turbine lifetime TL can be determined by simply scaling Eq. (37.2) appropriately. Thus, the expected amount of damage that will accumulate over a wind turbine’s lifetime is EŒD D TL TS Z Vout Vin Z 1 0 nS.sjTS; u/ N.s/ fU.u/dsdu; (37.3) where Vin and Vout are the cut-in and cut-out speeds, respectively, and fU.u/ is a PDF describing the distribution of mean wind speeds (typically a Weibull distribution [11]).
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