5 Probabilistic Fatigue Damage Assessment of Coastal Slender Bridges under Coupled
5.5 Results and Analysis
5.5.2 Probabilistic Analysis
The probabilistic modeling of the equivalent fatigue damage accumulation is illustrated using the stochastic load models proposed in the earlier sections. In the present study, a total number of 300 training samples are adopted to construct the five SVR models, i.e., 60 samples for each SVR model that corresponds to one specific vehicle type. Taking the V6 truck as an example, 60 training samples (denoted as U60) are first generated with the UDS scheme, which are uniformly distributed in the design space determined by the four influential parameters, i.e., GVW, Vw, Hs and Tp. After performing the necessary
FEAs 60 times, the SVR model can be established for approximating the response surface between the input loading parameters and the output daily equivalent fatigue damage accumulation. The SVR model of the U-rib butt joint under the V6 truck, wind and wave loads is shown in Fig. 5.10. Fig. 5.10(a) shows the response surface of the GVW and Vw when the wave loading parameters are supposed as Hs=2.0 and Tp=6.5s.
It is observed that the response surface is nonlinear, i.e., the D increases nonlinearly with the Vw and GVW.
The D first increases slowly when both GVW and Vw are small and then increases quickly as both GVW
and Vw go higher. In addition, the extreme wind alone loading condition can induce significant fatigue
damage accumulation, e.g., the D can reach 2.0×10-5 under wind speed of 25 m/s, which is comparable to that due to heavy truck load alone when GVW=600 kN. However, for even heavier truck load such that GVW ≥800 kN, the additional wind speed does not amplify the D significantly.
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(a) (b)
(c)
Figure 5.10 Response surfaces of the daily equivalent fatigue damage accumulation D for the U-rib butt joint under the V6 truck, wind and wave loads with U60 training samples: (a) GVW and Vw (Hs=2.0 m,
Tp=6.5s); (b) Vw and Hs (GVW=100 kN, Tp=6.5s); and (c) Hs and Tp (Vw=2m/s, GVW= 50 kN)
In order to study the effects of the coupled wind and wave loads on the daily equivalent fatigue damage accumulation, the response surface of Vw and Hs is also presented by assuming GVW=100 kN and Tp=6.5s,
as shown in Fig. 5.10(b). It is also found that the D increases slowly when both the Vw and Hs are small,
and then increases much faster with higher Vw and Hs. It is also worth mentioning that the extreme wave
alone can introduce large fatigue damage accumulation, i.e, D reaches 2.1×10-5 under H
s =6.0m.
Furthermore, Fig. 5.10(c) also illustrates the response surface of the two wave loading parameters Hs and
Tp, when the wind speed and truck load are assumed as Vw=2 m/s and GVW= 50 kN. Different from the Hs
that large wave heights induce large fatigue damage accumulation, it is interesting to see the D does not vary too much for small Hs. For large Hs, however, small Tp can cause relatively large D in general.
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To quantitatively evaluate the prediction performance of the SVR model, three performance indices, i.e., root mean square error (RMSE), mean absolute error (MAE) and mean absolute percentage error (MAPE), are utilized [254]. These three commonly used performance indices are proposed to measure the deviation between the predicted and observed (refers to FEA results in the present study) values, and smaller indices usually indicate better prediction performance. These three performance indices are defined as,
2 1 ˆ RMSE N i i i y y N
(5.10) 1 1 ˆ MAE iN yi yi N
(5.11) 1 ˆ 1 MAPE N i i 100% i i y y N y
(5.12) where yi and yˆi (i=1~N) are the ith observed and predicted outputs and N is the number of predicted data.It is noted that multivariate inputs are involved in the SVR model. For more clear and thorough presentation of the evaluation results in 3D domain, the validation case study takes two parameters as variables while the remaining two are assumed to be deterministic. Without loss of generality, the input parameters GVW (V6 vehicle) and Vw are treated as variables, and the input parameters Hs and Tp are
assigned with constant values, i.e., Hs=2.0 m, Tp=6.5s. To further investigate the influence of the number
of training samples on the prediction performance of the SVR model, three different sets of training samples that consist of 30 samples (U30), 60 samples (U60), and 90 samples (U90), respectively, are adopted for comparison purpose. Subsequently, a new testing sample comprised of 60 samples is employed to evaluate the three established SVR models, in which the corresponding error measures are tabulated in Table 5.6. It is observed that compared with the SVR models with U60 and U90 training samples, the SVR model with U30 training samples generates the largest statistical errors in terms of all the three indices, indicating that the SVR model with U30 data has the worst performance. The reason is due to that 30 training samples are too sparse in the entire design domain to construct a SVR model with desirable prediction accuracy. In contrast, both U60 and U90 training samples are sufficient as the corresponding SVR models can generate
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much lower statistical errors. Since the SVR models with U60 and U90 training samples have similar good performance, as all the three performance indices for both models are very low and close, U60 is selected in the present study to save computational cost. Fig. 5.11 displays the fitted SVR model as well as the training and testing samples computed from the FEAs. It is observed that the SVR response surface is close to all the samples with maximum absolute difference less than 1.3%, showing that the nonlinear characteristics of the output daily equivalent fatigue damage accumulation is well captured by the SVR model.
Table 5.6 Performance evaluations of SVR models (V6 vehicle) with three different sets of training
samples
SVR model RMSE (×10-7) MAE (×10-7) MAPE (%)
U30 6.33 3.59 2.58
U60 4.46 2.58 0.86
U90 4.45 2.58 0.84
Figure 5.11 Fitted SVR model (for V6 vehicle) and the FEA-based results
After a total number of 300 FEA simulations, five SVR models for five types of vehicles are established and utilized for probabilistic modeling of the daily equivalent fatigue damage accumulation, D,
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with a large number of samples obtained from the established stochastic truck, wind and wave load models. It is found from the preliminary sensitivity analysis that the PDF of the predicted D with 1 million samples is almost identical to that with 5 million samples, and therefore, the sampling number is selected as 1 million to provide accurate distribution of the predicted D. The PDF of the predicted daily equivalent fatigue damage in U-rib butt joint is show in Fig. 5.12, which is fitted by three proposed distribution models. It is found that the Weibull distribution fits best, which has a maximum log-likelihood value in the three models. The estimated Weibull parameters for all the three welded joints are shown in Table 5.7. Compared with the conventional MCS that requires a large number of samples for the FEA simulations that are extremely time consuming, the proposed numerical framework provides a more efficient way for probabilistic modeling of the daily equivalent fatigue damage accumulation.
0.0 0.5 1.0 1.5 2.0 2.5 0 2 4 6 8 10 ×104 Pro b abil ity d ensity D (×10-5) Simulated data Weibull distribution Normal distribution Lognormal distribution
Figure 5.12 PDF of daily equivalent fatigue damage accumulation in U-rib butt joint Table 5.7 Parameters in Weibull distributions of D for three welded joints Welded joint Scale parameter λ (×10-5) Shape parameter k
U-rib butt joint 1.00 2.35
Rib-to-diaphragm joint 0.82 2.27
Rib-to-deck joint 0.73 2.32