on the safety of high-speed trains running on bridges during earthquakes Qing ZENG Elias G. DIMITRAKOPOULOS

Transcription

1 on the safety of high-speed trains running on bridges during earthquakes Qing ZENG Elias G. DIMITRAKOPOULOS the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

2 outline motivation proposed model results frequent earthquakes results rare earthquakes conclusions 2 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

3 outline motivation proposed model results frequent earthquakes results rare earthquakes conclusions 3 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

4 motivation contemporary high-speed railways (HSR s) Beijing-Harbin Beijing-Hong Kong Qingdao-Taiyuan Xuzhou-Lanzhou Shanghai-Chengdu Beijing- Shanghai China s HSR operational speed: 3-35 km/h total HSR length: 16, km (to 28/12/214) longest bridge : 164 km (over 4 bridges) bridge/line ratio : Legend Beijing-Shanghai, China 8.5% high-speed railway lines east-west lines north-south lines Shanghai-Kunming Shanghai-Shenzhen Taipei-Kaohsiung, Taiwan 73% Beijing-Tianjin, China 86.6% 4 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

5 u Bv (mm) u Br (mm) u Bv (mm) u Br (mm) VBI: bridge resonance and cancellation ten identical 3D vehicles single span curved simple bridge u: displacement; a: acceleration V: vehicle, B: bridge v: vertical, r: radial motivation v N V =1 L B =32 m a Vv a Vr u Bv u Br R=5 m (a) 5 v = 314 km/h (b) v = 458 km/h.5 5 vertical resonance (c) 1 vertical cancellation v = 268 km/h vertical resonance vert. cancellation -.5 radial resonance (d) radial cancellation v = 391 km/h (e) (f) 1st car the 42nd Risk,.2 Hazard & Uncertainty 5th Workshop, car June 216 Hydra Island, Greece

6 motivation accidents: derailment of trains during earthquakes Chūetsu Earthquake, Japan (M w = Oct 24) Jiashian earthquake, southern Taiwan (M w = Mar 21) Shinkansen railway, 2 km/h broke the 48-year safety record 3 km/h high-speed train from VBI to SVBI: Seismic Vehicle-Bridge Interaction the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 6

7 outline motivation proposed model results frequent earthquakes results rare earthquakes conclusions 7 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

8 proposed SVBI model railway bridges modelling vehicle bridge interaction approach multibody dynamics finite element method nonsmooth dynamics (214) the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 8

9 vehicle modelling brief history (a) (a) (a) (b) (b) (b) moving loads moving masses current trend beyond moving load analsis vehicle-bridge interaction analysis YB Yang (1995) interdisciplinary civil + mechanical engineering (c) (c) (c) moving sprung masses car body bogie wheelset the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 9

10 vehicle modelling railway train vehicle ti (a) X ti car-body Z ti X ti car-body (pitching) (pitching) (vertical) (vertical) (longitudinal) (b) (longitudinal) speed v (rolling) (lateral) speed v (vertical) Z ti Y ti (b) multibody assembly (vertical) (rolling) Z ti wheelsets, bogies, (lateral) car-body rigid bodies Y ti (c) bogie bogie wheelset wheelset (yawing) (longitudinal) Y ti X ti (yawing) Y ti X ti (vertical) (vertical) (longitudinal) (d) terminology z (vertical) ψ(yawing) y (lateral) θ (pitching) s (longitudinal) φ (rolling) suspension springs + dashpots e.g. 38 total DOF s (d) terminology z (vertical) ψ(yawing) y (lateral) θ (pitching) s (longitudinal) φ (rolling) the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 1

11 bridge subsystem equation of motion M u C u K u W λ W λ F B B B B B B B B B N N H T wheel speed v contact point i Y ti X ti expressed in the inertia global system direction matrices W B N and W B T e.g. for a beam element linear shape functions for the axial (longitudinal) and torsional DOF s cubic (Hermitian) shape functions for the flexural DOF s i-1 th beam element s i =vt ξ ith beam element L i i+1 th beam element the only non-zero entries in W B N and W B T elements in contact with the wheel as vehicle wheel moves, different bridge element contact with wheel W B N and W B T : time-dependent the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 11

12 vehicle modelling railway train vehicle f V 1 =.54 Hz f V 2 =.78 Hz f V 3 =.9 Hz multibody assembly wheelsets, bogies, car-body rigid bodies (a) first lateral-rolling mode (b) second lateral-rolling mode (c) vertical mode suspension springs + dashpots f V 4 = 1.43 Hz (d) yawing mode f V 5 = 1.6 Hz (e) pitching mode 12 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

13 vehicle modelling vehicle-dynamics Y O Z space-fixed system I s i =vt R ti X R i Y ti O ti R ir O ir z ir Y ir y ir Z ti ψ ir X ti X ir φ ir trajectory system TI θ ir body-fixed system IR Z ir 3 systems of reference (Shabana 21) inertial (space-fixed) system I body-fixed system IR moving trajectory system TI moving trajectory system TI (for curved paths) moves along the curve longitudinal direction O ti X ti : tangent to the curve origin and orientation : defined by the arc length, s i the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 13

14 vehicle subsystem equation of motion on a curved path Z O IrOO G Z G O G Y X Y G space-fixed (inertial) system I ground acceleration I roo G X G Ir O G O ir ground-fixed system IG IrOO ir Ir O G O ti s i =vt Y ir θ ir Y ti O ir z ir y ir Z ir ψ ir Z ti O ti X ir φ ir moving trajectory system TI r I O ti O ir X ti body-fixed system IR T T t t m t t t M L L H I H i i i i i i i I I IR IR IR T T F m L t γ H t I γ ω I ω i i i i i i i i i i v I I R IR IR IR IR IR IR t i i i T G m I I OO F L r G 14 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

15 vehicle subsystem equation of motion M () t u C u K u W λ W λ F V V V V V V V V V N N T T expressed in the moving trajectory system M V (t) time varying for curved path λ N and λ T = normal and tangential contact force vector W V N and W V T = direction matrices of λ N and λ T (a) F c cosϕ ϕ+ϕ h F c λ N2 F G cosϕ F G sinϕ λ T1 F G λ N1 F V = gravity, inertia (centrifugal and coriolis forces) due to the curved path seismic loads if considered λ N2 λ T1 λ N1 ϕ Δh the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 15

16 coupled vehicle-bridge system equation of motion Mu Cu Ku Wλ F contact forces (coupling) (a) F c cosϕ ϕ+ϕ h F c λ N2 λ N2 F G cosϕ F G sinϕ λ T1 F G λ N1 λ T1 λ N1 Y W contains Z the coupling is time-dependent e 2 (b) e 1 y H global matrices contact forces λ t V V M C M,, B C B M C V V V K u F K,, =, B u B FΔh B K u F ϕ λ N W WN WH, λ, h λt V V W T WN WT bridge deck, B WN B WT t WN t 16 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

17 vertical [mm] coupled vehicle-bridge system wheel-rail contact interaction Mu Cu Ku Wλ F point and direction of contact continuous contact the wheel is in contact with the rail nonlinear wheel, rail profile r L δ L λ T8 λ N8 rail d L wheelset λ N7 λ T7 δ R r R -2-2 thread rail head flange -5 normal direction tangential direction wheel/rail lateral [mm] the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

18 coupled vehicle-bridge system wheel-rail contact interaction Mu Cu Ku Wλ F point and direction of contact wheel-rail separation/ detachment (uplifting) 5 g N is the minimum distance single/double separation the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

19 coupled vehicle-bridge system wheel-rail contact interaction Mu Cu Ku Wλ F normal direction (c) continuous normal contact contact model vs detachment (separation) λ N - kinematic constraint nonsmooth dynamics continuous contact contact-detachment transition is formulated as a Linear Complimentary Problem (LCP) nonsmooth approach contact: zero contact acceleration detachment: zero normal contact force detachement g N g N continuous contact g N N detachment N impact and recontact Newton s law normal contact displacement g N = and (g N < ) velocity jump on the velocity level the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

20 coupled vehicle-bridge system wheel-rail contact interaction Mu Cu Ku Wλ F tangential direction creep lateral / longitudinal forces creep spin moment rolling contact λ Ty1 λ Mz1 y ir λ Tx1 θ ir (lateral) z ir (vertical) ψ ir (yawing) O ir s i φ ir (rolling) λ Mz2 λ Tx2 Y ti Z ti (pitching) X ti λ Ty2 O ti the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

21 coupled vehicle-bridge system wheel-rail contact interaction tangential direction creep lateral / longitudinal forces creep spin moment Mu Cu Ku Wλ F nonlinear relationship between creep forces and creepage λ Ty1 λ Mz1 y ir λ Tx1 θ ir (lateral) z ir (vertical) ψ ir (yawing) O ir s i φ ir (rolling) λ Mz2 λ Tx2 Y ti Z ti (pitching) X ti λ Ty2 O ti wheel-rail friction model the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

22 coupled vehicle-bridge system final equations of motion t t t t t t t M u C u K u F * * * * C E W G W M C C 2vW G W * 1 T T K E W NG NN WN M K v WNG NN WN * 1 T F E WNG NN WN M F F v WG NN rcn * 1 T 1 1 T N NN N N NN N effect of: normal contact forces creep forces rail irregularities M * time-dependent (for curved path) K * and C * time-dependent and coupled F * time-dependent E identity matrix; G(t) -1 = (W T M -1 W) -1 the effective mass during contact ()' and ()'' the derivatives with respect to the arc length 22 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

23 car body coupled Y vehicle-bridge system ti Z ti final equations of motion bogie sources of excitation t t t t t t t M u C u K u F * * * * wheelset λ N5 external λ N3 excitation: λ N1 earthquake ground motion, wind loading etc. al) λ N5 λ N3 λ N1 6 λ T3 5 wing) x rolling) Coriolis and centrifugal forces: curved bridges and/or curved rails 4 2 λ N8 λ T4 self-excitation 3 : wheel-hunting, 1 creep forces, rail irregularities (c) Node 1 s i = vt λ T2 contact point i speed v λ T1 elevation irregularities r cn N c ith element Node 2 r 2 L i A cos( s ) i c n n n n 1 Z ti Y ti X ti (d) 2l a φ h r w λ N7 y H wheelset rail A H y A 2 vt sin LH H H H v 23 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

24 proposed analysis platform finite element software multibody software in-house algorithm pre-processing ANSYS bridge FEM model (M B, C B and K B ) Workbench vehicle multibody model (M V (t), C V and K V ) processing system global matrices (mass/stiffness/damping) M*, C* and K* post-processing deformed shapes, member forces diagrams ANSYS bridge Workbench vehicle time-history solutions perform VBI analysis in MATLAB time-history solutions the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 24

25 outline motivation proposed model results frequent earthquakes results rare earthquakes conclusions 25 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

26 case studies straight continuous highway bridge interacting with trucks during earthq. (2D) A1 P1 P2 A2 straight continuous slab highway bridge interacting with trucks no earthq. (3D) curved continuous railway bridge interacting with train during earthq. (3D) straight steel arch truss railway bridge interacting with train no earthq. (3D) straight cable-stayed bridge interacting with train no earthq. (3D) the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 26

27 horizontally curved continuous bridge SVBI frequent earthquakes (a) A 36 m P1 45 m P2 45 m P3 45 m P4 36 m A m Z X m m 13.8 m (b) O O Y X R = 75 m railway double track the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 27 27

28 horizontally curved continuous bridge deformation animation of the whole bridge 1 passing vehicles, without earthquake SVBI frequent earthquakes top view elevation view the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 28

29 horizontally curved continuous bridge deformation animation of the whole bridge SVBI frequent earthquakes 1 passing vehicles, with earthquake top view elevation view the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 29

30 horizontally curved continuous bridge SVBI frequent earthquakes deformation animation of the vehicle 1 passing vehicles, with earthquake elevation view 3D view front view the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 3

31 a Vv (m/s 2 ) a Vr (m/s 2 ) a Vv (m/s 2 ) a Vr (m/s 2 ) a Vv (m/s 2 ) u Bv (mm) u Bv (mm) u Br (mm) u Br (mm) u Bv (mm) horizontally curved continuous bridge vertical and radial disp. of bridge midpoint VBI model no earthquake excitation VBI model u: displacement no earthquake a: acceleration excitation (a) u: displacement a: acceleration.2 (a).2 effective -.2 uncracked effective uncracked SVBI frequent earthquakes vertical and radial accel. of vehicle car-body 1 vehicles R=75 m (b) a Vv (c).3 (d) first 6vehicle first 4vehicle (c).1.3 (d).1.3 first vehicle -.1 first vehicle effective stiffness of piers effective -.3stiffness of piers time (s) -.2 riding time (s) comfort time (s) effective stiffness of piers effective of the stiffness passengers of piers acceleration 1 2 3of the 4 car-body time (s) time (s) a Vr (b) VBI model no earthquake excitation u: displacement a: acceleration v (a) = 12 km/h u Bv ua.2 Br Vv a Vr L=27 m 1 vehicles R=75 m v = 12 km/h u Bv u Br effective L=27 m uncracked (c).3 first veh effective stiffness of pie vertical 2. m/s 2 radial (lateral) 1.5 m/s the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

32 u Br (mm) u Br (mm) FS FS u Br (mm) u Bv (mm) u Bv (mm) FS FS u Bv (mm) horizontally curved continuous bridge.3q k,1 : the additional -2 mass due to traffic SVBI frequent earthquakes no passing vehicles earthquake excitation in 3 directions u: displacement no passing vehicles FS: Fourier Spectrum no passing vehicles u Bv earthquake excitation in 3 directions u Br earthquake excitation in 3 directions (a) L=27 m u: displacement Z2 u: displacement Y Z FS: Fourier Spectrum ground acc. u max : 1.33 mm X R=75 m Y FS: Fourier Spectrum ground acc. X (a) 2 (b) 4 x 1-5 (a) without.3q k,1 u max : 1.33 mm 2 without.3q (b) without.3q k,1 k,1 with.3q k,1 4 u max : 1.33 mm u min : with Q mm k, Hz u min : mm -1 (c) u min : mm (c) x (d) (c) u max : 9.54 mm Hz (d) u max : 9.54 mm without time.3q(s) k,1 u min : -9.3 mm u min : -9.3 mm time (s) frequency (Hz) time (s) 32 vertical and radial disp. of bridge midpoint the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

33 a Vr (m/s a Vr 2 ) (m/s 2 ) a Vr (m/s 2 ) a Vr (m/s 2 ) a Vr (m/s 2 ) a Vv (m/s a Vv 2 ) (m/s 2 ) a Vv (m/s 2 ) a Vv (m/s 2 ) a Vv (m/s 2 ) horizontally curved continuous bridge vertical radial vertical and radial accel. of car-body VBI model earthquake excitation in 3 directions a: acceleration VBI V-1 model and V-1: the first and tenth vehicle earthquake excitation in 3 directions (a) a: acceleration 2 V-1 running V-1 passing V-1 and V-1: on the the first bridge and tenth vehicle over the bridge accel. of the vehicle on bridge > accel. of the vehicle on ground SVBI frequent earthquakes 1 st vehicle (a) 1 s 2 V-1 running 6.82 V-1 s s passing s on the bridge over the bridge 1 s 6.82 s -1 strong s 5 s shaking V-1 - vertical -2-1 strong 2 4 shaking s 5 s (c) V-1 - vertical -22 V-1 running V-1 passing 2 on the 4 bridge 6 8 over 1 the 12 bridge (c) 21 s V-1 running 6.82 V-1 s passing on the bridge s over the bridge s 1 s 6.82 s strong -1 shaking s 5 s strong -1 V-1 - radial -2 shaking s 2 5 s V-1 - radial -2 time (s) time (s) VBI model earthquake excitation in 3 directions a: acceleration V-1 and V-1: the first and tenth vehicle a Vv a Vr v = 12 km/h 1 vehicles (a) 2 V-1 running V-1 passing on the bridge over the bridge a Vv a Vr Z L=27 m 1 1 s Y vehicles 6.82 s ground acc. X v = R=75 km/hm s s (b) Z L=27 m 2 V-1 running Y 1 th V-1 running on the ground vehicle on the bridge ground acc. -1X R=75 m strong (b) 1 2 V-1 s running s s 5 s 6.75 shaking V-1 s running s s on the ground on the bridge V-1 - vertical s 6.75 s s (c) strong -1 2 V-1 running V-1 passing s 5 on shaking s the bridge over the bridge strong V-1 - vertical s 6.82 s 2 shaking s s s 5 s (d) V-1 - vertical -2 2 V-1 running V-1 running 2 on the 4 ground 6 8 strong on 1 the 12 bridge (d) V-1 s running 6.75 shaking V-1 s running s s on the ground s 5 s on the bridge s V-1 - radial -2 1 s s s strong time (s) -1 shaking s strong 5 s V-1 - radial -1-2 shaking s 25 s V-1 - radial -2 time (s) time (s) the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

34 horizontally curved continuous bridge SVBI frequent earthquakes the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 34

35 horizontally curved continuous bridge SVBI frequent earthquakes vertical radial vertical and radial disp. of bridge midpoint the conventional seismic analysis overestimates the response of the bridge the multibody vehicle acts as a additional damping to the bridge and reduce its response 35 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

36 horizontally curved continuous bridge SVBI frequent earthquakes 2 different earthquakes comfort: vertical and radial accel. of car-body safety: derailment and offload factor comfort: safety Derail factor Ti Ni Offloading factor Nli Nli Nri Nri 36 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

37 outline motivation proposed model results frequent earthquakes results rare earthquakes conclusions 37 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

38 derailment u factor Vr (mm) offload u VФ factor (mm rad) a Vr (m/s 2 ) zoom in a λn1 Vv (m/s (kn) 2 ) u Vr (mm) u VФ (mm rad) a Vr (m/s 2 ) a Vv (m/s 2 ) (a) horizontally curved continuous bridge (c) a Vr and a Vv : radial and vertical accel. of car body u1 Vr and u VФ max: mm : vertical and rolling disp. of car body λ N1 : normal contact force of wheel 1 P and Q: dynamic vertical and horizontal force min: 57.6 mm (a) (e) 5 2 max: 6.73 m/s 2 λn1 (kn) λn1 (kn) max: 6.73 m/s 2 5 SVBI rare earthquakes (d) 2 Z max: 17.6 mm rad ground λ Ty Y acc. Q λ N X contact P -2 angle δ min: mm rad (b) (f) 5 t = s t = s 2 detachment detachment t = s t = s 1-5 recontact max: 2.5 m/s 2 recontact (c) (d) 2 1 max: mm max: mm rad (g) (h) contact:.8.4 max:.41 nonzero max: normal.64 contact force detachment:.6 offload factor derailment factor = Q / P zero normal contact force min: 57.6 mm -2 = P - P.4 st / P st.2 min: mm rad -1 three components of the ground motion P st : static vertical force (e) recorded on April 25, 1992 in Cape Mendocino (f) t = s t = s 2 2 USA, at Petrolia 1 station 2 (.59g, 3.66g,.19g) detachment 1 detachment 2 3 time (s) time (s) t = s t = s 1 1 recontact recontact (b) in λn1 (kn) -5 max: 2.5 m/s 2 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 38

39 separation sequence separation no need for high lateral accelerations SVBI rare earthquakes list list of of state state change change of of wheel wheel 11 t = t = s s 1 st 1 st detachment t = t = s s 1 st 1 st recontact t = t = s s 2 nd 2 nd detachment t = t = s s 2 nd 2 nd recontact (a) (a) wheel wheel (b) (b) t = t = s s 2 1 st 2 1 st detachment of of wheel wheel wheel wheel 11-4 wheel wheel (c) (c) t = t = s s 2 uplifting of wheel 1 contact 2 uplifting of wheel 1 2contact after 2 nd 2 after 2 nd detachment of of wheel wheel wheel wheel (d) (d) t = t = s s 2 2 nd 2 2 nd recontact of of wheel wheel ground Z Z ground acc. acc. Y Y X X t = t = s s initial initial condition wheel wheel flange contact 2 flange contact of of wheel wheel wheel wheel contact 2 contact of of wheel wheel wheel wheel 11-4 wheel wheel left left wheel/rail coordinate right right wheel/rail coordinate wheel wheel profile profile rail rail profile profile tangential vector vector normal normal vector vector 39 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

40 derailment factor offload factor u Vwr (mm) u Vr (mm) u VФ (mm rad) a Vr (m/s 2 ) a Vv (m/s 2 ) SVBI rare earthquakes separation sequence a Vr and a Vv : radial and vertical accel. of car body u Vr and u VФ : vertical and rolling disp. of car body u Vwr : radial disp. of the wheelset λ N1 : normal contact force of wheel 1 P and Q: dynamic vertical and horizontal force (a) 1 (b) 1 ground acc. Z Y X λ Ty Q contact P angle δ λ N derailment max: 9.38 m/s 2 max: 7.88 m/s (c) (d) 3 max: 14.1 mm rad 2 max: 211. mm 1 min: 46.6 mm min: mm rad -1-4 (e) (f) wheelset 4 derailment wheel 8 (wheels 7 and 8) 2 max: kn 2 max: 5 mm 1-2 (g) (h) max: 1.19 offload factor max:.89 1 = P - P st / P st derailment factor = Q / P.5 P.5 st : static vertical force (i) time (s) 5 (j) 5 wheel 7 wheel 8 rail left wheel/rail coordinate (mm) λn1 (kn) time (s) rail right wheel/rail coordinate (mm) 4 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

41 outline motivation proposed model results frequent earthquakes results rare earthquakes conclusions 41 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece

42 conclusions scheme for seismic response of interacting vehicle and horizontally curved bridges the mass / stiffness / damping matrix of the coupled vehicle-bridge system become time-dependent the scheme accommodates easily different number of vehicles, and with various DOFs the seismic response of the bridge affects (adversely) the safety of the vehicle Vehicles-bridge-earthquake timing problem lack of performance (comfort and safety) criteria for vehicles probabilistic complicated dynamics, multi-parametric problem the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 42

43 list of relevant journal publications references 1. Zeng Q., Dimitrakopoulos Ε.G. (216) Seismic Response Analysis of an Interacting Curved Bridge-Train System Under Frequent Earthquakes Earthquake Engineering and Structural Dynamics, published online, 13 Jan 216 DOI 1.12/eqe Zeng Q., Yang Y.B., Dimitrakopoulos Ε.G. (216) Dynamic response of high speed vehicles and sustaining curved bridges under conditions of resonance Engineering Structures, published 1 May 216, vol. 114 pp 61-74, DOI: 1.116/j.engstruct Dimitrakopoulos E.G. & Zeng Q. (215) "A Three-dimensional Dynamic Analysis Scheme for the Interaction between Trains and Curved Railway Bridges" Computers & Structures, vol Paraskeva T.S., Dimitrakopoulos Ε.G., Zeng Q., (216) Dynamic Vehicle-Bridge Interaction Under Simultaneous Vertical Earthquake Excitation Bulletin of Earthquake Engineering, accepted 28/5/216 the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 43

44 thank you for your kind attention! the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 44

45 low-cost high impact bamboo bridges the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 45

46 steel arch truss bridge case studies and results Ting Sihe Bridge of High-speed railway line in China (215) the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 46

47 steel arch truss bridge case studies and results deformed shape of the whole bridge 1 vehicle: speed of train = 3 km/h the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 47 47

48 steel arch truss bridge deformation animation of the vehicle case studies and results elevation view 1 vehicle: speed of train = 3 km/h vibration due to the track irregularities 3D view front view the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 48 48

49 steel arch truss bridge case studies and results deformed shape of the whole bridge 1 vehicle: speed of train = 3 km/h the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 49 49

50 case studies and results cable-stayed bridge with the passage of MTR Kap Shui Mun Bridge (KSMB) in Hong Kong the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 5

51 u Bt (rad) a Vh (m/s 2 ) a Vh (m/s 2 ) u Bv u Bh (mm) u Bv u Bh (mm) u Bv u Bh (mm) u Bv u Bh (mm) (b) a Vv (m/s 2 ) u Bt (rad) x 1-5 horizontal and torsional displacements ten 3D vehicles a Vv L = 75-1 m 1 vehicles v = 144 km/h a u Bv accelerations Vh -2 (a) (a)(c) u Bh u Bt (d) (b) 1 1 x (b) a Vv (m/s 2 ) u Bt (rad) (c) u Bh dimensionless time (vt/l) u Bh case studies and results u Bv -1 cable-stayed bridge with the passage of MTR (a) and (b) bridge midpoint vertical (c) and (d) vehicle vertical and horizontal torsional horizontal bridge displacement u Bv (b) ten 3D vehicles L = 75 m v = 144 km/h (c) -1 vertical -1 a Vv (m/s 2 ) u Bt (rad) (c) (d) x u Bh u Bt dimensionless time (vt/l) the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece u Bh vehicle acceleration u Bv vertical x 1-5 horizontal a Vh a Vv u Bh u Bv ten 3D vehicles L = 75 m v = 144 km/h 1 vehicles (a) (b) u Bv kpa 51 ve

52 case studies and results cable-stayed bridge with the passage of MTR the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 52

53 36.m 46.m straight highway bridge case studies and results A1 P1 P2 A2 63.8m 118.6m 63.8m speed v speed v u B L=246.2m z ground accel. O x the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 53

54 case studies and results 5-span highway continuous bridge with the passage of trucks deformed shape of the whole bridge and the vehicle 1 vehicle: speed of truck = 8 km/h the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 54

55 case studies and results 5-span highway continuous bridge with the passage of trucks deformed shape of the whole bridge 5 vehicle: speed of truck = 8 km/h the 42nd Risk, Hazard & Uncertainty Workshop, June 216 Hydra Island, Greece 55