Tacoma Narrows Bridge

Transcription

1 Tacoma Narrows Bridge

2 Structure description: The bridge is a three-span suspension bridge (335m-853m-335m). The towers are 130 meters high. The deck consists of a section with steel beams of 2.45 m thickness, and 12 m width. Suspenders are placed every 33.5 m in the left and right spans, and every 42.5 m in the central span.

3 Bridge collapse: Tacoma Narrows Bridge. With a wind of 67 km/h, the bridge suffered a kind of aerodynamic inestability known as flutter. The Aeroelastic flutter phenomenon is generated by varying wind forces that change due to slight movements of the structure, affecting the stiffness and damping of the system. When the damping becomes negative, a small oscillatory movement is amplified exponentially until the structure collapses. The fluttering occurs when the energy transferred to the structure is such that the mechanical damping is unable to dissipate the energy. Once the critical fluttering speed is exceeded, the phenomenon increases and causes the collapse.

4 Bridge collapse : Tacoma Narrows Bridge.

5 Finite element model description: 3D modal and harmonic analysis are chosen, modeling the bridge with the following structural elements: Deck: Shell type with quadrangular elements and steel material. Longitudinal side beams of deck: Beam elements and steel material with rectangular section. Transversal beams deck: Beam elements with rectangular steel section. Towers: Beam elements with rectangular steel section. Catenary: Truss elements of steel material with circular section. Vertical suspension cables: Truss elements of steel material with circular section.

6 Finite element model description: Catenary and Vertical suspension cables -Truss Towers-Beam Deck-Shell Transversal beams Longitudinal Beams

7 Finite element model description: Geometry and catenary equations are as follows: Dimensions in inchs

8 Finite element model description: Catenary and suspender dimensions: Dimensions in inchs.

9 Finite element model description: a)thirty seven (37) transversal beams: b).- Hundred sixty (160) Longitudinal stiffeners:

10 Finite element model description: c)thirty seven (37) suspenders (truss elements): d).- Six (6) catenaries (truss elements):

11 Finite element model description: e).- Eight (8) longitudinal lateral beam stiffeners: f).- Three (3) decks (shell elements):

12 Finite element model description: g).- Four (4) cross member in towers (beam elements): h).- Eight (8) Pile Towers (beam elements):

13 Boundary conditions: 1.- Lateral spans, catenary and suspenders:

14 Boundary conditions: 2.- Towers and foundation:

15 Boundary conditions: 3.- Top towers and deck suspenders:

16 Input units: Tacoma Narrows Bridge. CivilFEM powered by Marc allows choosing between several units and evaluate each of the parameters in that particular unit:

17 Analysis and results: We want to obtain the natural frequencies of the bridge (modal analysis), and calculate the frequency that mobilizes the torsional mode. Then we will simulate the wind and vortex shedding that caused the bridge collapse by performing a harmonic analysis. As shown in the results table, the first torsion mode is number 7, with a frequency of Hz. Experimental data shows that the bridge collapsed at 0.2 Hz frequency. Modal analysis: Results shows that the deck bending and torsional modes are all very close

18 Analysis and results: -RESULTS Modal analysis-

19 Analysis and results: -Bending deck (mode 4)- -Torsional deck (mode 7)- f 4 = 0,141 Hz f 7 = 0,195 Hz The torsion frequency is not 1.5 times the bending frequency (f 7 / f 4 < 1,5): flutter risk.

20 Analysis and results (Mode 7):

21 Harmonic Analysis: CivilFEM powered by Marc allows the importation of a model (or different elements): The user can choose what to import.

22 Harmonic Analysis: _ Damping considered: 1 %. _ Harmonic frequency range = 0 to 1 Hz. _ Number of substeps = 100. _ Check frequency versus vertical displacement on node 111.

23 Harmonic Analysis: Vorticity is simulated using positive and negative vertical harmonic loads on the deck (unitary load), that produce the oscillation of the bridge:

24 Harmonic Analysis: It is verified that the maximum amplification occurs at a frequency of 0,192 Hz:

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