Distributed computing for the evaluation of the aeroelastic response and sensitivity analysis of flutter speed of the Messina Bridge
F. Nieto, S. Hernández & J. Á. Jurado School of Civil Engineering, University of La Coruña, Spain
Abstract
This paper explains the formulation that allows the calculation of a bridge flutter speed and the evaluation of the sensitivity of the aeroelastic response with regards to the mechanical parameters of the deck. This methodology requires the previous evaluation of the natural frequencies and mode shapes of the bridge in second order theory, and the sensitivity analysis of those frequencies and mode shapes. Then, the advantages of introducing distributed computing in order to save time and share the computational effort between a set of connected computers are presented. Finally, the results obtained using the previously introduced methodology and distributed computing are shown for the Messina strait bridge. Keywords: distributed computing, aeroelasticity, flutter, sensitivity analysis, Messina Bridge.
1 Introduction
In addition to the inherent difficulties arising in the design and construction of cable supported bridges, the length of the span covered by this class of structures has impressively increased in the last decades up to the 1991 m of the Akashi strait bridge in Japan. The ambitious project of suspension bridge for the Messina strait bridge is already in its definitive step and other challenging proposals as the Tsugaru Strait Bridge in Japan [1] or the Rial Altas crossing [2] in Spain are under study. For these long span bridges environmental solicitations as wind effects are often more significant than service loads. Therefore a lot of expertise is necessary in the design of such impressive structures.
Fluid Structure Interaction and Moving Boundary Problems 465
In order to determine the aeroelastic response of long span bridges, the engineers have nowadays two powerful technical tools: sensitivity analysis of the flutter speed and parallel programming that allows the time required for the task to be decreased.
2 Hybrid method for aeroelastic analysis
The method develops in two phases, the first one, which is experimental, is carried out in small wind tunnels, where reduced model of a segment of the deck is tested under wind flow. The aim of this step is to identify the flutter coefficients * * *, ,i i iA H P (i = 1,…6) which are used to obtain the wind forces on the deck from its displacements and velocities [3]. The relationship of wind forces on the deck L: lift, D: drag and M: moment, and deck displacements v, w, ϕx and velocities xwv ϕ,, is
2 * 2 * 2 *
4 6 32 2 * 2 * 2 *
6 4 32 * 2 * 2 2 *
6 4 3
* * 2 *1 5 2
2 * * 2 *5 1 2
2 * 2 * 3 *5 1 2
12
12
x
x
D K P K P BK P vL U K H K H BK H wM BK A BK A B K A
BKP U BKP U B KP U vU BKH U BKH U B KH U w
B KA U B KA U B KA U
ρϕ
ρϕ
− − = − +
− − − + −
−
. (1)
Eqn (1) can be presented in matrix notation [4] as
uCuKP aa += (2) Second step in this approach is to produce a finite element model of the bridge and carry out a dynamic analysis using the wind loads defined in eqn (2). The dynamic equilibrium can be written as
uCuKKuuCuM aa +=++ (3) by using modal descomposition eqn (3) is finally transformed into an eigenvalue problem.
( ) 0wIA =− µµ (4) Eqn (4) provides a set of pairs of conjugate complex eigenvalues ( )n1,iµi = , being n the number of modes included in the analysis.
-i i i i i ii iµ α β µ α β= = + (5)
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Real part iα is related to the modal damping. Therefore positive damping values are required for the structure to be stable under wind flow. If iα becomes zero for a given mode iα under a specific wind speed, then flutter instability can arise and such speed fU is the maximum speed the bridge can withstand. Therefore obtaining flutter speed fU is done by solving the eigenvalue problem indicated in (4) according to the flowchart of figure 1a) and then, proceeding iteratively by increasing values of wind speed Uf until obtaining zero value for a given real part iα . Such procedure is shown in figure 1b).
µ α + βj = ij j j m=1,...,
j = 1
j m =
β jp
[ ( ) ] = A I w 0β − µ jp µ
l m= 1,...,
β md
dif jp l l= min| | β − β
dif < tol
α βj j i
[ ( ) ( ) ] = A I w 0β − α + β j j ji µ
Que cumplen
j j= + 1
β βjp md =
NO
NO
SI
SI
β βj md=
α αj md =
α − βl l iα + βl l i
µ α − βj = ij j
[ ( ) ( ) ] = A I w 0β − α − β j j j µ
U
j = 1
( − µ ) =A I w 0µ
µ α + βj = ij jj m=1,...,
ξ ξmin = min aj
con ξ α α βaj j = - / ( + )
2 1/2j j2
ξ min 0
Flameo =
= U Uffω βmin
U U U = + ∆µ α − βj = ij j
SI
NO
Figure 1: a) Flow diagram of eigenvalue problem. b) Flow diagram of flutter speed.
3 Sensitivity analysis of flutter speed
During the design process of a cable supported bridge engineers modificate properties of the structure based upon intuition and expertise backed on
Satisfying
YES
YES
YES
Flutter
a) b)
with
_
_
Fluid Structure Interaction and Moving Boundary Problems 467
experimental testing and computer based structural analysis in a trial and error procedure. Instead of that, a more rational way of improving intermediate prototypes is the use of sensitivity analysis. This is a scientific term coined in the field of structural optimization that defines the derivatives of structural responses with respect to a design variable [5-6]. Therefore, a study aimed to obtain the sensitivity analysis of flutter speed Uf with regard to a design variable, can be carried out and the derivative dUf /dx is obtained. This study does not require supplementary flutter analysis and thus is more convenient than the trial and error techniques. Sensitivities are very useful for designers. For instance if 0fdU dx > it means that increasing the design variable x, flutter speed will be higher while the behaviour will be reverse if negative. The magnitude of the value informs of how much variation we can expect to happen. Therefore at each design step sensitivity analysis show the engineer which design values are worthy of being modified (those having great absolute values) and which must be the direction of the change. Sensitivity analysis techniques have been used since many years ago in car, aviation and space industry and sooner or later they will find their way in bridge engineering. At flutter wind speed and considering that αi = 0 and βi = Kf Uf/B where Kf is the reduced frequency, expression (4) can be written as
f fK Ui
B µ
+ =
A I w 0 . (6)
The approach for obtaining the sensitivity of flutter speed starts by derivating this expression with regards to a design variable x
f fK Ud idx B µ
+ =
A I w 0 , (7)
or
f f f ff f
dK dU K Ud dd i U K idx dx B dx dx B dx
µ µµ µ
+ + + + =
w wA w A I w I 0 . (8)
Premultiplying by T
µv that is a vector accomplishing
( )jiµ β =v A I 0+ , (9)
it turns out
468 Fluid Structure Interaction and Moving Boundary Problems
Knowing the sensitivity analysis of the natural eigenresponse of the structure [7] the following complex numbers can be defined:
T T TAx AU AK
f f
h h hx U Kµ µ µ µ µ µ
∂ ∂ ∂= = =
∂ ∂ ∂A A Av w v w v w . (11)
and
( ) ( ) f fT TU AU K AK
iK iUg h g h
B Bµ µ µ µ= + = +v Iw v Iw . (12)
The following equation can be written
f fU K Ax
d U d Kg g h
dx dx+ = − , (13)
where dxdU f / and fdK dx are real numbers. Multiplying by the conjugate
complex numbers kg and Ug the sensitivities of the flutter speed Uf and the reduced frequency Kf with regards to any design variable x can be finally written as
( )( )
( )( )
Im Im
Im Imf fU Ax K Ax
U K K U
d K d Ug h g hdx g g dx g g
− −= = . (14)
4 Distributed programming
The methodology previously described includes different tasks that are extremely computer time demanding [8]. Flutter analysis requires the previous evaluation of natural frequencies and mode shapes and computer calculation of aeroelastic eigenresponses solving a nonlinear problem. Sensitivity analysis of flutter speed also demands a lot of computer time and has to be done for each design variable selected. Evaluation of natural frequencies and mode shapes is a serial process, however evaluation of aeroelastic eigenresponses and sensitivity analysis can be carried out in parallel.
Fluid Structure Interaction and Moving Boundary Problems 469
This fact opens a wide room for the use of distributed computing by the utilization of clusters of personal computers that carry out the tasks broadcasted by a front-end CPU. Nowadays clusters of personal computers are not expensive and are at hand in any engineering company or research center.
5 Application example: Messina Bridge
The Messina strait bridge will connect soon Sicily and Reggio-Calbria in Italy. Its 3300 m long central span will become this bridge in the longest cable-supported bridge in the world. In figure 2 a virtual view of the proposed bridge is shown.
Figure 2: Virtual view of the proposed Messina Bridge.
A three-dimensional finite element model has been made by this research team considering the three boxes that configure the bridge deck. In the figure 3 a picture of the deck section is shown while figure 4 shows an image of the structural model made by the authors.
Figure 3: Messina Bridge deck
Identification of the sensitivity analysis of flutter speed of the Messina Bridge requires to independent consideration of the mechanical parameters of lateral boxes, which are identical, and the central one. This research is going on and in this paper only the results produced by considering the mechanical parameters of central box, namely Iy, Iz, J, and A are described.
470 Fluid Structure Interaction and Moving Boundary Problems
The mechanical parameters of the central box of the bridge deck considered in the present paper are listed in table 1.
Table 1: Mechanical parameters of the Messina Bridge central deck.
Iy (m4) Iz (m4) J (m4) A (m2) Central deck 0.301 2.12 0.738 0.39
Figure 4: Messina Bridge finite element model.
Table 2: Natural eigenresponse of the Messina Bridge.
nº type freq (rad/s) freq (Hz) Period (s) 1 LS 0.2004 0.0319 31.35 2 VA 0.3682 0.0586 17.06 3 LA 0.4018 0.0639 15.64 4 VS 0.4838 0.0770 12.99 5 TA 0.5390 0.0858 11.66 6 LTS 0.5623 0.0895 11.17 7 C 0.5871 0.0934 10.70 8 C 0.6028 0.0959 10.42 9 LTS 0.6140 0.0977 10.23
10 VS 0.6422 0.1022 9.78 11 LTA 0.6637 0.1056 9.47 12 LS 0.6907 0.1099 9.10 13 VA 0.7642 0.1216 8.22 14 LTS 0.7850 0.1249 8.00 15 LA 0.7974 0.1269 7.88 16 C 0.8308 0.1322 7.56 17 LON 0.8478 0.1349 7.41 18 LA 0.9030 0.1437 6.96 19 LTA 0.9181 0.1461 6.84 20 LTA 0.9308 0.1481 6.75
5.1 Natural eigenresponses
Up to twenty modes were included in the flutter analysis and the natural frequencies of such modes appear en table 2 along with the type of displacement associated. The abbreviations of the words used in table 2 are, L: lateral, V:
Fluid Structure Interaction and Moving Boundary Problems 471
vertical, T: torsion, LON: longitudinal, C: cable, S: symmetric and A: asymmetric. In table 3 the natural frequencies obtained in this paper are compared with those provided by other researchers. It can be concluded that the results agree generally. The abbreviations of the words in table 3 are, PDM: Politecnico de Milano, YNU: Yokohama National University and UDC: University of La Coruña.
Table 3: Comparison of natural frequencies among modeling.
Mode PDM YNU UDC
Horizontal (1º sym.) 1 0.033 1 0.031 1 0.032
Horizontal (1º asyim.) 2 0.059 2 0.059 3 0.064
Vertical (1º asym.) 3 0.061 3 0.064 2 0.059
Vertical (1º sym.) 4 0.08 5 0.078 4 0.077
Torsion (1º asym.) 5 0.081 4 0.076 5 0.086
Horizontal (2º sym.) 6 0.084 6 0.084 6 0.089
Torsion (1º sym.) 9 0.097 8 0.093 9 0.098
Vertical (2º sym.) 12 0.107 11 0.104 10 0.102
Vertical (2º asym.) 14 0.128 15 0.123 13 0.122
Torsion (2º sym.) 15 0.129 14 0.123 14 0.125
Table 4: Sensitivity analysis of the natural frequencies of the Messina Bridge.
In table 4 the derivatives of the natural frequencies of the Messina Bridge with regards to the design variables are shown. The design variables considered in the
472 Fluid Structure Interaction and Moving Boundary Problems
present work have been: Iy (vertical bending inertia), Iz (lateral bending inertia), J (torsional inertia) and A (deck section area) of the central spine of the deck. Table 4 shows that the design variable that provides a greater change in the value of natural frequencies is the central box section area.
5.3 Aeroelastic response
Distributed programming has been used in order to obtain the aeroelastic response of the Messina bridge. The graphic outputs provided by the parallel program developed by the authors are shown in figures 5, 6 and 7. These figures show the evolution of the real part of the complex eigenvalues iα , of the imaginary part of the same eigenvalues iβ and the quotient /i iα β− considering 16 vibration modes. The results have been calculated taking into account 8 flutter coefficients (A1
*, A2*, A3
*, A4*, H1
*, H2*, H3
*, H4*), which haven been obtained
in the aerodynamic wind tunnel property of this research team. The obtained flutter speed has been Uf=86,768 m/s and the reduced frequency has been Kf=0.33995.
Figure 5: Evolution of the real part of the eigenvalues.
Figure 6: Evolution of the imaginary part of the eigenvalues.
Fluid Structure Interaction and Moving Boundary Problems 473
Figure 8: Numerical results of the Messina Bridge.
With the purpose of testing the flutter speed obtained in the present work a table is shown comparing the results obtained by other research teams. It must be taken into account that the structural model and the flutter coeffients used in this work are different from those used by Politecnico de Milano and Yokohama National University.
474 Fluid Structure Interaction and Moving Boundary Problems
Table 5: Flutter speeds obtained for Messina Bridge by different research teams.
PDM YNU UDC Velocidad de flameo (m/s) >80 71.8 86.768
5.4 Sensitivity analysis of the aeroelastic response
A very important issue is to test the analytical results obtained for the sensitivities of the flutter speed and the reduced frequency with regards to the design variables. In order to check the analytical results obtained in the present work, sensitivity analysis have been carried out using central finite differences (F.D.) with step sizes of 1% and 5%. The results are shown in table 6.
Table 6: Sensitivity analysis of the aeroelastic response of the Messina Bridge.
Long span suspension bridges are one of the most challenging structures in the world and therefore every kind of useful technology as sensitivity analysis or parallel programming should be used in their design process. Sensitivity analysis of flutter speed has been presented in this paper as a powerful tool for identifying the sense of variations and the amount of the expected change for given values of the mechanical parameters of the deck. Aware of the computer time demanded by the presented methodology, the authors promote the use of distributed computing by arranging clusters of personal computers that are nowadays very cost effective. Differences amongst numerical values of flutter speed make clear that given the significance of this bridge more research is needed to identify its aeroelastic behaviour
Acknowledgement
This research has been funded by the Spanish Ministry of Education and Science under project BIA2004-01898.
References
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[2] Hernández, S., The Rias Altas link. A challenging crossing, in Strait Crossing 2001, pp. 407-414, J. Krokebas (ed.) Balkema, 2001.
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[6] Hafka, R.T., Gürdal, Z. & Kamat, M.P. Elements of Structural of Optimization, Kluwer Academic Publishers, 1990.
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