As discussed earlier, the multi-scale FEA, i.e., the global-local modeling is adopted for fatigue-related stress analysis. The global-local modeling aims to capture the accurate stress responses at critical welded joints through independent local refined FE model with 3-D shell/solid elements, whereas the boundary conditions are adopted from the corresponding global less-refined FE model. The global-local FE modeling
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approach has been used in many applications including the fatigue analysis of long-span bridges, which confirms the effectiveness for large-scale simulation [80,250,251]. The procedure for the multi-scale FEA of the OSD is as follows. Firstly, the global structural dynamic analysis of the bridge under loading samples created by UDS (see Fig. 5.6) is performed based on a coupled vehicle-bridge-wind-wave (VBWW) analytical platform developed by the authors [198]. The global dynamic analysis takes into account the complex interactions among the bridge, traffic, wind and wave. Next, a separate local model is developed using refined shell elements (see Fig. 5.7) with detailed geometry. It is noted that the length of the local refined model should be simulated to have more than four times of its height, to avoid the end effect according to the Saint-Venant’s Principle [252]. After applying both the boundary conditions and loading conditions extracted from the global VBWW analysis, the stress time history at the structural details can be obtained.
Based on the simulation results from the global analysis, the deck segment around the bridge mid-span is identified as the critical region. Hence, a local model of 18m OSD at the bridge mid-span is built using refined Shell63 element with detailed geometry, as shown in Fig. 5.7. The segment length of the local refined model is 6 times of the diaphragm spacing in order to eliminate the potential influence of the rigid region on the cutting boundary based on Saint-Venant’s principle. The dimensions of the cross section and the U-rib are shown in Fig. 5.8. Since a majority of heavy-loaded trucks are travelling on the slow lane as indicated in Table 5.1 and Fig. 5.2, the structural members under the slow lane is more likely prone to potential severe fatigue damage than those under the other two lanes. As a result, the present study will focus only on the structural members and details near the slow lane. In addition, since the pavement is not included in the FE model, a spreading angle of 45° for a vertical uniformly distributed wheel load is applied in the bridge deck. Considering the thickness of the pavement (5.5 cm) and the dimensions of the front and back wheels (contact area is 30cm×20cm and 60cm×20cm), the updated distribution load areas for the front and back wheels are calculated as 41cm×31cm and 71cm×31cm.
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(a)
(b)
Figure 5.7 Local FE model: (a) 18m steel deck segment; and (b) refined portion of the slow lane between the two diaphragms
Firstly, we will discuss the results based on the multi-scale FEA under one representative loading case from the UDS samples. The loading parameters are: GVW=920 kN (V6 truck), Vw=15.6 m/s, Hs=3.7 m,
Tp=7.4 s. As discussed earlier, both the wind and wave loads are applied laterally on the bridge without
considering the wind and wave directions. The linear wave model is used to generate the random waves using the shallow water TMA spectrum for a given Hs and Tp [253]. The spectral representation method is
adopted to simulate the stochastic wind fluctuations using the Kaimal spectrum and Lumley-Panofsky spectrum [26]. The time histories of wind and wave are used to calculate the structural loads applied on the bridge [198]. The traffic flow for each type of vehicle is simulated in accordance with the vehicle statistics as discussed previously. In addition, a constant vehicle speed of 20 m/s is adopted for all the simulations.
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Figure 5.8 Dimensions of OSD: (a) cross section; and (b) U-rib with typical fatigue-prone details Fig. 5.9 compares the stress time histories for the U-rib butt joint under different combinations of the loading parameters, in order to clearly show their contributions to the stress histories. As shown in Figs. 5.9(a) ~ (c), the stress response due to the coupled wind and wave loads is larger than those caused by individual wind load or wave load. To observe the stress time history from the coupled VBWW dynamic analysis, the stress response including two V6 truck passages is shown in Fig. 5.9(d). By comparing the Figs. 5.9(c) and (d), it is observed that when the additional truck loads are included, the stress level has been increased dramatically, i.e., the stress peak value increases from 11.9 MPa under coupled wind and wave loads to 62.9 MPa for coupled VBWW system. In addition, the stress response due to truck load only is also shown in Fig. 5.9(d) for comparison. It is shown in Fig. 5.9(d) that both the amplitude and the shape of the vehicle-induced stress time history are affected when the additional wind and waves are included in the analysis. For example, the stress peak value increases from 51.6 MPa under vehicle only loading scenario to 62.9 MPa for coupled VBWW system by 21.8%. For trucks with smaller GVWs, the wind and wave loads are found to have more profound influences on the vehicle-induced stress responses. The above observations indicate that all the external loads from the traffic, wind and wave can contribute to the
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structural dynamic responses and it is necessary to include the coupled VBWW analysis for the fatigue evaluation. -12 0 12 -12 0 12 100 150 200 250 -12 0 12 S tre ss ( MP a) Wind only S tre ss ( MP a) Wave only (c) (a) (b) S tre ss ( MP a) Time (s)
Coupled wind and wave
5.0 5.5 6.0 6.5 7.0 7.5 8.0 8.5 -10 0 10 20 30 40 50 60 70 Two V 6 truck passages (d) Stress (MP a) Time (s) Vehicle only VBWW 62.9 51.6
Figure 5.9 Stress time history segments of butt joint of U-rib: (a) wind only; (b) wave only; (c) coupled wind and wave; (d) vehicle only and coupled vehicle-bridge-wind-wave
After the stress time histories are obtained, the fatigue stress ranges and the corresponding number of cycles can be calculated using the rain-flow counting method. Table 5.5 summarizes the equivalent stress range Sre and the number of stress cycles nt in 10-minute duration with 10-V6 truck passages. Sre and nt
calculated from the vehicle alone, wind alone, wave alone loading scenarios are also listed for comparison. It is observed that for all the three welded joints, the truck loads prone to induce large Sre with relative small
nt, whereas the wind and wave loads are likely to cause small Sre with relative large nt. Since each individual
load has different effects on the Sre and nt, the equivalent fatigue damage accumulation D has a better
indication for the structural fatigue damage which combines the two essential parameters into one. By further comparing the D under each loading scenario, it is clear to quantify the contribution from each load to the structural fatigue damage. In this case study in particular, the truck load contributes most to the fatigue damage accumulation while the wave load contributes the least. The equivalent fatigue damage accumulation is utilized in the subsequent analysis.
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Table 5.5 Summary of Sre, nt and D in 10-minute duration with 10-V6 truck passages (Sre: MPa)
Loading scenario
U-rib but joint Rib-to-diaphragm joint Rib-to-deck joint
Sre nt D Sre nt D Sre nt D
VBWW 26.47 240 6.17E-06 34.21 194 5.39E-06 28.76 228 3.77E-06 Vehicle alone 40.61 55 5.11E-06 54.30 43 4.78E-06 37.56 76 2.80E-06 Wind alone 13.32 210 6.88E-07 15.52 158 4.10E-07 16.24 202 6.01E-07 Wave alone 10.57 162 2.65E-07 11.83 102 1.17E-07 11.89 183 2.14E-07