8 A Framework for Developing Efficient Vehicle-Bridge Interaction Models Within a Commercial Finite Element Software 71 built-in eigenvalue extraction methods: Lanczos, automated multi-level substructuring (AMS), and subspace iteration [11]. The Lanczos solver is used in this study. αR =ξ 2ω1ω2 ω1 +ω2 (8.3) βR =ξ 2 ω1 +ω2 (8.4) 8.3 VBI Verification Study In this study, two VBI models, a quarter-car model and a half-car model, are created using the proposed approach in Abaqus, and their results are compared to the dynamic response of hard-coded VBI models taken from the literature [1, 7]. For the quarter-car model, the following properties taken from Yang et al. [7] are used to model the beam: length L =25 m, Eb =2.75 ×10 10 N/m2, μb =4800 kg/m, Ib =0.12 m4, and no damping. The vehicle properties are Mq =1200 kg, Kq =500,000 N/m, V =10 m/s, and zero damping. The natural frequencies of the bridge (ωb) and vehicle (ωv) can be computed individually using the built-in frequency step in Abaqus and are ωb =2.08 Hz (first mode’s natural frequency) and ωv =3.25 Hz. A surface profile was not defined during the quarter-car simulation. For the half-car model, the following properties are used from Locke [1] to model the beam: L = 21.3 m, Eb =200 ×10 9 N/m2, μb =5600 kg/m, ξ =3%, and Ib =0.0842m4. The properties of the vehicle are Mv =12,404 kg, Iv =172,160 kg/m2, MFU =725.4kg, MRU =725.4kg, KFS =727,812 N/m, KRS =1,969,034 N/m, KFU =1,972,900 N/m, KRU =4,735,000 N/m, V =10m/s, a =3m, and b =3 m. Similar to the quarter-car model, the natural frequencies are computed using the frequency step and are ωb =6.00 Hz (first mode’s natural frequency), ωv1 =1.27 Hz (pitch with front unsprung bounce), ωv2 =2.17 Hz (pitch mode with rear unsprung mass bounce), ωv3 =9.75 Hz (front unsprung masses bounce), and ωv4 =15.4 Hz (mode rear unsprung bounce). In both models, the beams are discretized into 50 elements following a trial-and-error Scheme. A road profile is generated using the power spectral density (PSD) method defined by ISO-8608 standards [19] where Road Class A was used with a displacement PSD (Gd) of 32 ×10−6 m−1 with a spatial frequency (n0) of 0.1 cycles/m. The spatial frequency band spans from 0.001 to 10 cycles/m at an increment of 1/L. During the implicit dynamic step, a common issue that users face when employing contact algorithms in Abaqus is contact chatter, which is when a slave node falls off a master surface [16]. To overcome this issue, the moderate dissipation application is used to stabilize the model and reduce contact chatter [16] where α=−0.41421, β =0.5, and γ =0.91421, while the time step used was 0.001 seconds. A sensitivity analysis was carried out to test different numerical damping values and found the recommended use of the moderate dissipation application removed contact chatter with and resulted in negligible change in vehicle response and change in bridge response. Figure 8.4 shows the results that were extracted from the quarter-car simulation for both the proposed FE model and Yang’s FE model [2], where Fig. 8.4a, b are the acceleration signals extracted at midspan of the bridge and center of mass 0 0.5 1 1.5 2 2.5 Time on Bridge (sec) -0.04 -0.02 0 0.02 0.04 Acceleration (m/s 2 ) Yang FE Abaqus FE 0 0.5 1 1.5 2 2.5 Time on Bridge (sec) -0.04 -0.02 0 0.02 0.04 Acceleration (m/s 2 ) Yang FE Abaqus FE a b Fig. 8.4 Quarter-Car dynamic response curves. (a) Acceleration signal of bridge midspan. (b) Acceleration signal of sprung mass
RkJQdWJsaXNoZXIy MTMzNzEzMQ==