### 1. Introduction

*m*-order spectral moment using the slope

*m*of the S-N curve. However, since it cannot reflect the interaction between low-frequency and high-frequency modes, the accuracy of fatigue damage is low.

### 2. Data Processing

### 2.1 Generation of Response Spectrum

*α*

_{1}and

*α*

_{2}are within the range of 0.2–0.9 and 0.1–0.8, respectively (Park et al., 2014). The spectral bandwidth parameter

*α*and the

_{n}*n*th order spectral moment

*m*are defined as follows.

_{n}*=*Wave frequency

*S*(ω)

_{a}*=*Response spectrum of stress amplitude

*a*

### 2.2 Data Extraction of Stress Time History

*ω*=

_{i}*ω*−

_{i}*ω*

_{i}_{− 1},

*R*(

*ω*) is the response spectrum, ω

_{i}*is the angular frequency,*

_{i}*ϕ*is the phase angle,

_{i}*t*is time, and

*n*is the number of frequencies. The angular frequency and phase angle are generated by a random number giving an equal distribution within the allowable range. In the process of extracting the time history of stress data, the stresses with the shortest period should be taken into account so as not to miss them. Therefore, it is important to select a reasonable and appropriate time increment in order to accurately capture the maximum and minimum of the stress during the IFFT process.

### 2.3 Rain-Flow Counting Process

### 3. Step Study of Approximate Spectral Moment

### 3.1 Definition of Rain-Flow Counting Moment

*MRR*) through numerical simulation and then to obtain an approximate spectral moment that is close to the rain-flow counting moment. For the response spectrum, the rain-flow counting moment is defined as follows (Dirlik, 1985):

^{RFC}### 3.2 Existing Approximate Spectral Moment

### 3.3 Step 1: Combination of Bandwidth Parameter

*b*and

*c*in the bandwidth parameter is assumed as follows.

*b*and

*c*using the bandwidth parameter, the range of exponents is determined while excluding the range of low accuracy of the R-squared values through parametric studies:

*MRR*(

^{RFC}*q*) obtained from numerical simulation. To show the difference between

*MRR*(

^{RFC}*q*) and

*b*and

*c*increase, the R-squared values of the first spectral moment gradually decrease. On the other hand, the R-squared values of the remaining spectral moment increase until

*b*is 1.1 and

*c*is 0.9. However, it decreases after that. As for the first-order spectral moment, it can be seen that the R-squared value is maximized when

*b*is 0.4 and

*c*is 0.9.

### 3.4 Step 2: Combination of Special Parameter

*MRR*(

^{RFC}*q*) by combining only the bandwidth parameters, a special parameter was used (Jun and Park, 2020): where,

*m*= spectral moment

*k*= 0.01 to 2.5, 0.01 interval

*α*. An iterative numerical calculation is performed in consideration of the following three conditions, and the R-squared values are confirmed through regression analysis between the special parameter and

_{k}*MRR*(

^{RFC}*q*) using the rain-flow counting moment obtained from the numerical simulation. In this study, the moment

*MRR*(

^{RFC}*q*). As shown in the table, excluding the fifth spectral moment, the accuracy of the remaining spectral moments is 99.7% or more. From this result, it is confirmed that the consideration of the special parameter is very appropriate to obtain the approximate spectral moment. However, as the order of the spectral moment is increased, the simple combination formula of the special parameter becomes complicated, so it is difficult to apply it to the actual design.

### 3.5 Step 3: Combination of a Special Parameter and Exponential Bandwidth Parameters

*μ*and spectral moment

_{k}*MRR*(

^{RFC}*q*) obtained from the numerical simulation, the R-squared values between

*MRR*(

^{RFC}*q*) and

^{st}and 2

^{nd}-order spectral moment, the maximum R-squared values are shown when the coefficient

*k*is 0.01. As the coefficient k increases, the R-squared values of the spectral moment decrease. This means that it is more advantageous not to consider the special parameter for the 1

^{st}and 2

^{nd}-order spectral moments. When the coefficient

*k*is 0.6, the R-squared values of the remaining spectral moments show the maximum values and give more accurate results than the approximate spectral moment

*k*is greater than 0.6, the R-squared values tend to decrease. From the results of step 3, the approximate spectral moments are summarized below:

### 3.6 Step 4: Combination of an Exponential Special Parameter and Exponential Bandwidth Parameters

*a*of the special parameter, the final range below is determined while excluding the range with low accuracy of the R-squared values through a parametric study:

*MRR*(

^{RFC}*q*) obtained from the numerical simulation, the R-squared values that show the difference between

*MRR*(

^{RFC}*q*) and

*a*increases, the R-squared values of the first spectral moment decrease. On the other hand, the R-squared values of the remaining spectral moment show a tendency of increasing as the exponent

*a*increases and then decreasing as it passes a specific value. As for the spectral moments from the first-order to the fifth-order, it can be seen that the R-squared value is maximized when the exponent

*a*is −1.0, −0.2, 0.4, and 0.7.

### 4. Comparison

### 4.1 Four Candidate Formulas

*MRR*(

^{RFC}*q*) and the highest equivalent approximate moment

### 4.2 Comparison of Candidate Formulas

*X*Relative distance error of

_{i}=*i*spectrum

^{th}*N =*Total number of spectra

### 5. Conclusions

Whereas other studies considered only the 2

^{nd}or 3^{rd}-order approximate spectral moments, this study contributed to increasing the accuracy of fatigue damage assessment by extending them to 4^{th}–order terms or higher.The previous models considered the approximate spectral moment with the spectral moment

*m*_{0}–*m*_{4}or the linear combination of the bandwidth parameters*α*_{1}and*α*_{2}. In this study, a special parameter or exponential special parameter was combined with the bandwidth parameter through a step-by-step study to develop several empirical formulas.In addition, simplicity and convenience were considered for an actual engineering application, and stability and accuracy of the fatigue analysis solution can be enhanced by applying approximate spectral moments with the R-squared values of more than 97% comparing with rain-flow counting moments.