Derailment of High Speed Trains Moving over Bridges under Earthquakes

Transcription

1 Derailment of High Speed Trains Moving over Bridges under Earthquakes 1 Y. B. Yang and 2 Y. S. Wu 1 President, YunTech / Distinguished Prof., NTU 2 Sinotech Engineering Consultants, Inc. 4 th Kuang Hwa Forum, Tongji University December 11, 2011

2 Outlines Solution method Concept of contact force Train-rails-bridge system Earthquake considerations YQ ratio of a wheel Instability criterion Derailment risk plot Concluding remarks 2

3 Concept of contact forces Vehicle Body Upper part Truck Suspension System k DOFs Others... Wheel part H n v wn v H wn 1 n 1 H 2 v w2 H 1 v w1 v cn v cn 1... v c2 v c1 V n V n 1 V 2 V 1 3

4 Vehicle equations of motion Vehicle DOFs Upper or non-contact part: Wheel or contact part: Vehicle equations Force vector: Contact force: Constraint eqn: 4

5 Upper-part displacements Partitioned eqn: First row: Newmark s procedure Displ incremments: 5

6 Solve for contact forces From second row: Introducing the constraint eqn, Each contact force: 6

7 Summary of solution procedure Discretize the upper part of the vehicle equations by Newmark s β method Solve for the upper-part displ. {d u } Substitute {d u } into the wheel part of the vehicle equations Obtain the contact forces {f c } Solve the bridge equations by treating the contact forces {f c } as consistent nodal forces 7

8 Train-rails-bridge system left track section central track section right track section train rail LSR element L bridge ballast RSR element CFR element lateral ground motion vertical ground motion bridge element 8

9 Problem of concern Two sources of vibration Train movement + earthquake 3D vehicle model Car body, bogie, and Wheelsets Horizontal and vertical forces Structural model Bridge and embankment Rails and track Earthquake characteristics Acceleration + velocity + displacement 9

10 C Body Bogie Rail Wheelset Ballast V7 V 5 V 3 V 1 Sleeper Bridge ( or Soil Roadbed) ( a) Sleeper Rails Track B 2nd wheelset H H 6 H H 2 1 Track A 4th wheelset H 7 H 5 3rd wheelset ( b) H 3 H 1 1st wheelset C Body Sleeper Ballast Rails Track B Wheelset H V 8 8 Track A Bogie H 7 V7 Bridge () c 10

11 Car body Car modeling Vertical, lateral, rolling, yawing, & pitching 5 degrees of freedom Front and rear bogies Each has 5 DOFs 4 wheelsets Each has 3 DOFs: vertical, lateral & rolling Entire car: 27 DOFs 11

12 Rail irregularity elevation irregualrity ( vertical profile) r v ideal geometry 4 types of irregularity gauge irregularity Elevation standard half gauge r g ideal geometry Alignment r h alignment irregularity Superelevation Gauge irregularity superelevation irregularity ( cross level) 2r c 12

13 Finite element simulation Train 1. Idealized as a series of identical vehicles. 2. Each vehicle : one car body, two bogies and four wheelsets (27 DOFs). Track 1. Ballast type. 2. Double tracks (Tracks A and B). 3. Simplified as a twin rail system lying on a single-layer ballast foundation. 4. Modeled by CFR, LSR and RSR elements. Bridge 1. Box-girder concrete bridge. 2. Modeled by 3D straight beam element 13

14 Rail elements (a) Central portion v r1 v r2 u r1 r2 u r 2 r1 1 2 c v k bv kbh bv b1 c bh v b2 u b1 u b 2 b1 b2 1 2 rail ballast bridge l (b) Left portion k bv c bv k bh c bh r2 v r2 2 u r2 rail ballast Right portion r1 v r1 1 u r1 k bv c bv k bh c bh rail ballast (c) 14

15 Equations of motion for rail and bridge elements considering effects of ground motion a. Central finite rail (CFR) element (Track A) where Equation of motion considering the ground motion where 15

16 b. Left semi-infinite rail (LSR) element (Track A) where f t c k Al fal fal fal c. Right semi-infinite rail (RSR) element (Track A) where 16

17 d. Bridge element (Track A & Track B) where and 17

18 Solution of vehicle-rail-bridge interaction equation Dynamic condensation technique Equation of motion for the vehicle in partitioned form Obtain the expression of contact force by using Newmark difference scheme and constraint relations where V i = vertical contact force, H i = lateral contact force, i = 1 ~ 8 for 8 wheels. 18

19 Form the condensed equation for the Vehicle-Rails Interaction (VRI) element by the concept of consistent nodal loads. Assemble all the rails, VRI and bridge elements to yield the condensed equation for the whole vehicle-rails-bridge interaction system. Solve the resulting equation by Newmark s ß method. 19

20 Earthquake considerations Horizontal + vertical excitations Near fault vs. far field El Centro, Northridge earthquakes Chi-Chi Earthquake (TAP003)-far field Chi-Chi Earthquake (TCU068)-near fault Normalized to have a lateral PGA = 80 gal Elastic structures

21 Contact forces between wheels and rails (El Centro) 21

22 Contact forces between wheels and rails (Northridge) 22

23 Contact forces between wheels and rails (TAP003) 23

24 Contact forces between wheels and rails (TCU068) 24

25 YQ ratio of a wheel Criteria for derailment YQ Y Q Y: lateral contact force of the wheel Q: vertical contact force of the wheel Other definitions are also available 25

26 Time history of max YQ ratio of 4 wheelsets of train car 26

27 Bridge response to moving train under TAP Bridge Midspan Vertical Displacement (m) Chi-Chi Earthquake TAP003 v=60 m/s (lateral (EW)) v=60 m/s (lateral (EW) + vertical) v=30 m/s (lateral (EW)) v=30 m/s (lateral (EW) + vertical) Nondimensional Time (vt/l ) 27

28 Car response to moving train under TAP003 1st Car Vertical Acceleration (m/s^2) Chi-Chi Earthquake TAP003 v=60 m/s (lateral (EW)) v=60 m/s (lateral (EW) + vertical) v=30 m/s (lateral (EW)) v=30 m/s (lateral (EW) + vertical) -0.6 Nondimensional Time (vt/l ) 28

29 Bridge response to moving train under TCU068 29

30 Car response to moving train under TCU068 30

31 YunTech & NTU YQ ratio for 1st wheelset of 1st car (El Centro) 31

32 YQ ratio for 1st wheelset of 1st car (Northridge) 32

33 YQ ratio for 1st wheelset of 1st car (TAP003) 2 Maximum YQ Ratio of Train lateral PGA = 25 gal lateral PGA = 80 gal safety limit = 1.5 Chi-Chi Earthquake TAP003 Train Speed = 200 km/h Nondimensional Time (vt/l ) 33

34 YQ ratio for 1st wheelset of 1st car (TCU068) 34

35 Seismic stability of a wheelset under earthquake Stability of a train is assessed by the ratio of the lateral contact force H to vertical contact force V of the two wheels R & L of a wheelset (YQ ratio): H R H L YQ= V V For a wheelset (Elkins & Carter 1993) R YQ 1.5 stable and free of derailment 1.5 < YQ 2.0 at a high risk of derailment 2.0 < YQ derailment L 35

36 Seismic stability of a train under earthquake For entire train, Max. YQ 1.5 safety 1.5 < Max. YQ 2.0 possible instability 2.0 < Max. YQ instability The criteria used are conservative. 36

37 YunTech & NTU Risk of derailment of train under El Centro earthquake Uni-directional 37

38 YunTech & NTU Risk of derailment of train under Northridge earthquake Uni-directional 38

39 YunTech & NTU Risk of derailment of train under TAP003 earthquake Uni-directional 39

40 Risk of derailment of train under TCU068 earthquake Uni-directional 40

41 Risk of derailment of train under TAP003 earthquake Bi-directional Chi-Chi Earthquake TAP003 Lateral+Vertical Lateral PGA (gal) Level 5 (80 gal) Level 4 (25 gal) possible instability safety Chi-Chi Earthquake TAP003 Lateral+Vertical Level 3 (8 gal) instability Train Speed (km/h) 41

42 Risk of derailment of train under TCU068 earthquake Bi-directional Chi-Chi Earthquake TCU068 Lateral+Vertical Lateral PGA (gal) Level 5 (80 gal) Level 4 (25 gal) safety possible instability instability Chi-Chi Earthquake TCU068 Lateral+Vertical Level 3 (8 gal) Train Speed (km/h) 42

43 Remarks Type of trains Type of earthquakes YQ ratio of wheels Risk assessment criteria Elastic structures Warning system? 43

44 References Yang, Y. B., and Wu, Y. S. (2001), A Versatile Element for Analysing Vehicle-Bridge Interaction Response, Eng. Struct., 23, Wu, Y. S., Yang, Y. B., and Yau, J. D., (2001), Three- Dimensional Analysis of Train-Rail-Bridge Interaction Problems, Vehicle System Dyn., 36(1), 2001, Yang, YB, and Wu, YS (2002), Dynamic Stability of Trains Moving over Bridges Shaken by Earthquakes, J. Sound & Vibr., 258(1), Wu, Y. S., and Yang, Y. B. (2003), Steady-State Response and Riding Comfort of Trains Moving over a Series of Simply Supported Bridges, Eng. Struct., 25(2),

45 Thank you for your attention