Global sensitivity analysis of fatigue life based on structural health monitoring

Fatigue cracks are one of the most common causes of structure failure. Structural health monitoring (SHM) has shown important practical significance in ensuring the safety of civil infrastructures by providing continuous, accurate, and reliable monitoring at low cost. Given that data provided by SHM involves various types of uncertainty, this study presents the interval-random hybrid model for fatigue life based on fatigue crack growth theory. By introducing global sensitivity indices, the influences of different variables on the variance of structural fatigue life are analyzed. To enhance computational efficiency, the method is proposed by combining the space partition with unscented transformation (SP-UT).


Introduction
In aerospace, bridges, and other important infrastructures intended for long-term service, repeated cyclic loads can result in fatigue crack growth at the location of existing initial cracks.These cracks then continue to expand, ultimately leading to fatigue damage.Fatigue and cracks are the main causes of structure failure [1].Real-time and accurate monitoring of structural fatigue cracks during operation is necessary [2].The massive data collected by the SHM system include numerous factors that impact structural fatigue life [3].A broad range of uncertainties including randomness and fuzziness exists and must be effectively characterized and described.Mechanical models can be used to accurately predict fatigue damage life and perform reliability sensitivity analyses.These measures can help inform decision-making and logistical maintenance planning to ensure optimal structural protection.The history of research on structural fatigue reliability is relatively short, and many studies have focused on constructing fatigue reliability models [4][5].Some studies introduced sensitivity analysis to optimize the design of structure fatigue life, which is not yet well developed [6][7], mainly focusing on local sensitivity analysis and lack of consideration for various mixed uncertainty situations.Variance-based sensitivity indices provide a global perspective on the output performance variance [8].In this paper, variance-based sensitivity indices are introduced to analyze the important measure of various engineering uncertainties for the fatigue life of structures.Considering the diverse and uncertain nature of data from the SHM system, the interval-random hybrid model for fatigue life based on fatigue crack growth theory is developed.In addition, the corresponding indices and efficient and robust algorithms are proposed.

Sobol's indices
Sobol's indices are the most classic and widely used indices to determine the average effect of input variables on the output response when they are fixed at a specific point.Numerous scholars have proposed variance-based importance measures, and this paper utilizes the commonly recognized Sobol's indices for research purposes.Sobol's indices are based on the high-dimensional model decomposition equation, because the structural limit state function is an n-dimensional random independent variable, and then Y and V Y can be obtained as follows [9]: , , , , , , 1 , 2 ,, 1 Based on the above decomposition equation, i S for the contribution of the input variable i X to V Y is obtained as: where V Y is the total variance of the model output; ¼ is the average reduction in the variance of the model output when the variable i X is fixed over its entire domain of distribution.If i S is larger, it means that the uncertainty in i X contributes more to the variance of the output.

Variance-based sensitivity indices for interval-random model
Variables with large data dispersion are defined as interval variables.The model contains random and interval variables , , which are independent random variables with probability density function . Thus, the output response M has interval-random mixed uncertainty, and the variance of M is , Based on the distance measure proposed by Liu [10], the impact of the variable on variance is positively correlated with the distance between the unconditional and conditional variance.
G reflect the size of the contribution of i X and i Y to the uncertainty of the output variance, respectively.
G can be obtained as follows: , ,

Efficient method for variance-based global sensitivity indices
Yun et al. [11] proposed a highly efficient method by combining space-partition (SP) and unscented transformation (UT) for solving variance-based global sensitivity indices.By reducing the nonlinearity in the response function by SP and utilizing the strong capability of the UT to explore the probability, all the indices for each order can be estimated with a set of UT samples.The computation process is as follows: Step 1: We divide the entire integration space into 1 ( 1, , ) probability by the density function f X x of the random variable X and apply the UT and optimization toolbox to solve for the unconditional variance of the output response , Step 2: M Gaussian points 1 ( , , ) and their corresponding weights f x of the one-dimensional input variable i X .
Step 3: We fix the variable i X at ( 1, , ) .We apply the UT and optimization toolbox to solve the conditional variance , ( 1 , , ) G can be estimated by Equation (4)   and Equation ( 6).
Based on the above computation process, the computational cost of the SP-UT method to obtain

Fatigue life sensitivity analysis based on SHM
Runyang Bridge is a super large bridge composed of a cable-stayed bridge and a suspension bridge.The suspension bridge has a main span of 1, 490 m.The bridge uses oorthotropic steel decks (OSDs) which are prone to fatigue cracking, especially at the welded connection points between the top plate and U-rib.Thus, developing the SHM system on Runyang Bridge is critical.Figure 1 shows that fiber grating strain sensors are employed to detect the strain at the top plate and the U-rib weld [12].
where ADTT is average daily load cycles; re S is daily equivalent stress range; C is fatigue property coefficient; m is the inverse slope of the log-log curves; n is bridge service year; D is the linear coefficient of traffic growth; 0 a and c a are initial and critical crack length, respectively; Y a is the dimensionless function, which can be expressed as: According to Deng et al. [13], re S and ADTT are usually treated as a random distribution to fit the probability density function based on the long-term monitoring data, as shown in Figure 2, which reveals that there are large errors in the fitting results.Thus, this paper sets re S and ADTT as interval variables with the basic change interval being [2.6,8.7]Mpa and [3053,9413] respectively.Other parameters are considered random variables and the detailed parameter information can be found in Table 1.The global sensitivity indices of random variables and interval variables for the fatigue life variance are given in Figure 3 and Figure 4, respectively.The fatigue property coefficient C is more important than the initial crack length 0 a , and the daily equivalent stress range re S is more crucial than average daily load cycles ADTT , which offers a theoretical foundation to guide actual engineering.
From Figure 3 and Figure 4, the results obtained by SP-UT coincide with those of the MCS method, indicating the accuracy of SP-UT in calculating the variance-based global sensitivity indices.Table 2 gives the computation number of SP-UT and MCS, which shows that SP-UT is much more efficient than MCS.

Conclusion
Drawing upon measured health monitoring data, this research presents a variance-based global sensitivity analysis for fatigue life based on crack growth theory.Given the challenge of accurately fitting uncertainty data obtained from health monitoring with stochastic probabilistic models, determining the magnitude or boundaries of their unknowns is comparatively straightforward.Thus, this paper proposes the interval-random variance-based global sensitivity indices and uses the SP-UT method to enhance computational efficiency.Furthermore, the significance of various variables fatigue life is analyzed to provide a theoretical foundation for practical engineering.

Figure 1 .
Figure 1.SHM system on Runyang Bridge.The fatigue life model based on crack growth theory has the following function:

Figure 2 .
Figure 2. Bin histograms and fitted PDFs of re S and ADTT [13].

Figure 3
Figure 3. M i V X G for fatigue life of random variables.

Figure 4
Figure 4. M j V Y G for fatigue life of interval variables.

Table 2 .
Comparison of the computation number between two methods.