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Advances in aeroelastic analyses of suspension and cable-stayed bridges
Journal of Wind Engineering and Industrial Aerodynamics 74—76 (1998) 73—90
Advances in aeroelastic analyses of suspension and cable-stayed bridges Allan Larsen* COWI Consulting Engineers and Planners AS, Parallelvej 15, DK-2800 Lyngby, Denmark
Abstract Cross-section shape is an important parameter for the wind response and aeroelastic stability of long span suspension and cable-stayed bridges. Numerical simulation methods have now been developed to a stage where assessment of the effect of practical cross-section shapes on bridge response is possible. The present paper reviews selected numerical simulations carried out for a long-span suspension bridge using finite difference and discrete vortex methods. Comparison of simulations to existing wind tunnel data is discussed. Further, the paper addresses the aerodynamics and structural response of four generic cross-section shapes developed from the well-known plate girder section of the first Tacoma Narrows Bridge. Finally a case study involving the wind response of a 400 m main span cable-stayed bridge is discussed. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Numerical methods; Bridge aerodynamics; Buffeting; Vortex shedding excitation; Aeroelastic instability
1. Introduction The engineering discipline of bridge-aerodynamics was born of the spray rising from the fall of the first Tacoma Narrows Bridge into the Puget Sound in 1940. In his monumental investigation of the bridge collapse, Farquharson [1] covered a wide range of technical aspects ranging from experimental techniques over aerodynamics and structural dynamics to guidelines for bridge design. Probably, the singlemost important finding of the Tacoma Narrows investigation was that vortex-shedding excitation and flutter instability of a complete suspension bridge could be accurately represented by a spring-supported section model of the deck structure. This important
* E-mail: [email protected]. 0167-6105/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 0 0 7 - 5
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result was later corroborated by wind tunnel investigations conducted in the United Kingdom in connection with design of a suspension bridge for crossing of the river Severn, Frazer and Scruton [2]. Further versatility was added to section model testing by Davenport [3], who advanced a method for assessment of buffeting response of complete bridge structures to turbulent wind based on aerodynamic section data. Needless to say, section model testing offers substantial savings relative to testing of full aeroelastic bridge section models. Structural analyses of bridges have moved from laboratory testing of physical models to computer-based finite element modelling. This development has allowed the designer to experiment with different structural systems and configurations without resorting to expensive and time-consuming physical testing. Aerodynamic analysis of bridges has not seen a similar development due to the complexity of the fluid dynamic phenomena involved, hence most aerodynamic analyses of bridge structures are still restricted by aerodynamic data obtained from wind tunnel testing. Numerical fluid dynamic models and computer capacity have developed over the past decade to a stage where the bridge designer may start to exploit these new techniques in actual design work in much the same way as physical section model tests. In particular, numerical simulations appear well suited for design studies of the effect of cross-section shape on bridge response to wind loading, thus presenting an efficient tool for weeding out inefficient cross-sections before embarking on confirmatory wind tunnel testing. The present paper will highlight some recent comparisons between wind tunnel section model results and numerical simulations. The main body of the paper will be devoted to a design study of the effect of cross-section shape on bridge response taking the first Tacoma Narrows Bridge as an example. Finally the paper will outline a numerical design study carried out for determination of the most favourable cross-section shape for a 400 m main span cable-stayed bridge.
2. Model tests and numerical simulations The wind design of the East Bridge was carried out in the time span 1989—1992 and was based on extensive wind tunnel section model testing. Since that time numerical methods have developed to the extent that two-dimensional aerodynamic section data may be calculated with acceptable accuracy for design studies. In order to illustrate this point, measured and calculated wind load coefficients for the girder cross-sections of the East Bridge suspended spans and approach spans, shown in Fig. 1, will be compared. 2.1. East Bridge suspended spans Steady-state wind loads for the cross-section of the East Bridge suspended spans have been reported by two different workers using different numerical simulation techniques. Kuroda [5] applied a gird-based finite-difference method (FDM) using the pseudo-compressibility technique for solution of the incompressible two-dimensional Navier—Stokes equations at Reynolds number Re"3]105 (based
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Fig. 1. Girder cross-sections of the East Bridge suspended spans and approach spans. Table 1 Comparison of steady-state wind load coefficients obtained from numerical simulations and wind tunnel testing of a 1 : 80 section model. East Bridge Suspended spans Method
C
FDM [5] DVM [6] Experiment
0.071 0.061 0.081
D0
(-) C
L0
(-)
!0.100 0.000 0.067
C
M0
(-)
0.025 0.027 0.028
dC /da (-/rad) L
dC /da (-/rad) M
6.48 4.13 4.37
1.15 1.15 1.17
on cross-section width B). Walther [6] applied the grid-free discrete vortex method (DVM) for solution of the two-dimensional vorticity equation representing the flow around the cross-section at Re"105. The wind loads reported are made nondimensional through division with the dynamic head 1oº2 and section width B: 2 ¸ M D , C " , C " . (1) C " L M 1oº2B2 D 1oº2B 1oº2B 2 2 2 Table 1 compares simulated wind load coefficients C , C , C at zero angle of D0 L0 M0 attack and lift and moment slopes dC /da, dC /da to experimental values obtained L M from wind tunnel testing of a 1 : 80 section model as reported by Larsen [4]. Satisfactory agreement between simulations and experiment is demonstrated for most of the coefficients with the exception of dC /da obtained from the FDM L simulations. This coefficient is 48% in excess of the experimental data. In comparing C values it shall be remembered that the physical section model was equipped with D0 light tubular railings and crash barriers, whereas the numerical geometry models only reproduced the gross trapezoidal cross-section shape. Simple calculations allowing each of the railing components to be exposed to the free stream wind speed yield a drag contribution of *C "0.023 which brings simulations and experiment in D0 better agreement. 2.2. East Bridge approach spans Similar to the cross-section of the suspended spans the approach spans (Fig. 2, right) have been subject to two-dimensional numerical flow simulations. Selvam
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A. Larsen/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 73–90 Table 2 Comparison of drag coefficient and Strouhal number obtained from numerical simulations and wind tunnel testing of a 1 : 80 scale section model. East Bridge Approach spans Method
C
FDM — LES [7] DVM [8] Experiment
0.187 0.179 0.190
D0
(-)
St (-) 0.166—0.199 0.167 0.170
et al. [7] applied a finite-difference method including a large eddy simulation (LES) model for prediction of the important drag loading and Strouhal number and vortex-shedding frequency. Larsen and Walther [8] report similar results obtained by means of the discrete vortex method. Table 2 offers a comparison between numerical simulations and experimental results of drag coefficient and Strouhal number: D fH C " , St" , D 1oº2B º 2
(2)
where H is cross-wind section depth (H"7.0 m) and f is vortex shedding frequency. As in the case of the suspended span cross-section the numerical simulations are in fair agreement with experimental data. Again railings and crash barriers were not included in the numerical models, hence slightly lower C values are to be expected D0 when comparing with the experiment. Further comparisons between experiment and numerical simulations are presented by Larsen and Walther [8] for the H-shaped cross-section of the first Tacoma Narrows Bridge and for a twin-box cross-section developed for a fixed link across the Straits of Gibraltar. These results are equally promising indicating that numerical simulations of flow around bridge girders are worth while in bridge design and retrofit studies.
3. Models for bridge response to wind Bridge response to wind is mainly governed by the aerodynamic properties of the girder cross-sections, structural parameters such as mass, mass moment of inertia, eigenfrequencies and damping and for buffeting response the turbulence properties of the wind field. The appendix offers a brief run down of mathematical models which may be used for a first-order assessment of the three most important types of wind-induced response: (1) along-wind buffeting response (drag direction), (2) vertical
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vortex-shedding excitation and (3) critical wind speed for onset of flutter. Besides being of practical use, the models illustrate that bridge buffeting, vortex-shedding excitation and flutter stability may be calculated, once drag coefficient C , root mean D0 square lift coefficient CRMS, Strouhal number St and aerodynamic derivatives H* , L 1..4 A* are available for a given bridge girder cross-section. A more complete descrip1..4 tion of the use of aerodynamic cross-section data in bridge response analyses is offered by Scanlan [12]. The following section will focus on the above-mentioned aerodynamic properties and simulation by the discrete vortex method.
4. Discrete vortex method for 2D bridge deck cross-sections A distinct feature of flow about bluff bodies, stationary or in time-dependent motion, is the shedding of vorticity in the wake which balances the change of fluid momentum along the body surface. The vorticity shed at an instant in time is convected downstream by the mean wind speed but continues to affect the aerodynamic loads on the body. A mathematical model for the flow around bluff bodies was developed within the framework of the discrete vortex method and programmed for computer by Walther [6]. The resulting numerical code DVMFLOW establishes a “grid-free” time-marching simulation of the vorticity equation well suited for simulation of 2D bluff body flows. An outline of the mathematical model and the simulated flow about a flat plate is presented by Walther and Larsen [9]. The input to DVMFLOW simulations is a boundary panel model of the cross-section contour. The output of DVMFLOW simulations is time progressions of surface pressures and section loads (drag, lift and moment). In addition, maps of the flow field (vector plots), vortex positions and streamlines at prescribed time steps are available. Steady-state wind load coefficients and Strouhal number are obtained from time averages and frequency analysis of simulated loads on stationary panel models. Aerodynamic derivatives are obtained from post-processing of simulated time series of forced harmonic motion as detailed by Larsen [10] in a numerical investigation of five generic bridge deck cross-sections tested in a wind tunnel by Scanlan and Tomko [11].
5. Five Bridge deck cross-sections — an example The lively wind response (the galloping) and the final collapse of the first Tacoma Narrows Bridge, Fig. 2, was established to be due to the aerodynamically unfavorable cross-section shape and lightness of the bridge structure. Although a number of investigations has pointed out that H-shaped cross-sections similar to the first Tacoma Narrows are undesirable from an aerodynamic point of view they remain attractive from an economic point of view as well as due to ease of fabrication. The present example will thus consider the aerodynamic effect of four simple modifications of the parent H-shaped cross-section.
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Fig. 2. First Tacoma Narrows Bridge. Girder deck cross-section, elevation and structural data applicable to the first asymmetric mode of vibration [1].
Fig. 3. Cross-section shapes considered in the present study.
5.1. Cross-sections investigated The five cross-sections investigated are shown in Fig. 3. The parent cross-section denoted H is a slightly simplified version of the first Tacoma Narrows deck omitting cross-girders and curbs but reproducing the longitudinal edge girders and floor slab. Section C (channel type) is obtained from the H section simply by adding a top plate. Section R (rectangular type) is obtained by adding a bottom-plate to the C section. The CE section (channel/edge) is obtained
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from the C section by adding triangular edge-fairings to the C section. Finally the B section (box type) is obtained by adding the triangular edge-fairings to the R section or closing the bottom of the CE section. All dimensions in Fig. 3 are referred to the width B of the parent H section. The circumference of each cross-section was subdivided into a total of 300 surface vortex panels. The discretisation allowed flow at Reynolds number Re"105 (Re"ºB/l) to be simulated. 5.2. Simulation of flow about stationary sections Drag coefficient C , root mean square lift coefficient CRMS and Strouhal number St L D0 are obtained from simulations of the flow about the five cross-sections fixed in space. An angle of attack of 0° of the wind flow was assumed (angle between flow direction and section chord). Each simulation was run for 30 non-dimensional time units ¹"tº/B where t is the time, º is the flow speed and B is the chord length. A non-dimensional time increment *¹"0.025 was adopted throughout the simulations. At each time step the cross-section surface pressure distribution was computed from the local flux of surface velocity. The section surface pressures were finally integrated along the contour to form time traces of the section drag D and lift ¸ forces. Lastly, the computed aerodynamic forces were expressed in non-dimensional form following Eq. (1). Fig. 4 shows an example of the simulated time traces of C and D0 C obtained for cross-section H. L The C trace displays initial very high values associated with the instantaneous D start up of the flow simulation. After an exponential decay the C trace settles around D a mean value C "0.28 after approximately 5 non-dimensional time units. D0 C "0.28 is in satisfactory agreement with C "0.29!0.30 reported by FarD0 D0 quharson [1]. The C trace develops very distinct oscillations with period ¹+1.7 L associated with formation of vortex roll up in the wake — the well-known von Ka´rma´n vortex street, Fig. 5. The wake pattern obtained from simulation of the flow
Fig. 4. Simulated time traces of drag coefficient C and lift coefficient C for cross-section H. D L
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Fig. 5. Formation of von Ka´rma´n vortex street in the wake of cross-section H.
Fig. 6. Formation of von Ka´rma´n vortex street in the wake of cross-section B.
Fig. 7. Simulated time traces of drag coefficient C and lift coefficient C for cross-section B. D L
around cross-section B is shown in Fig. 6, whereas simulated time traces of C and D C are shown in Fig. 7. A summary of aerodynamic data for all sections is given in L Table 1. From Table 3 it is noted that the closed-box section B displays better aerodynamic performance than the remaining cross-sections, i.e. lower C and CRMS. The parent D L cross-section H appears to yield the worst aerodynamic performance.
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A. Larsen/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 73–90 Table 3 Flow fields in the vicinity of the cross sections and C , CRMS and St values extracted D0 L Cross section geometry and flow patterns
C D0
CRMS L
St
0.28
0.37
0.11
0.23
0.33
0.11
0.23
0.24
0.09
0.16
0.34
0.09
0.11
0.17
0.13
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By comparing Figs. 4 and 7 it is noted that the drag loading and the oscillating lift have decreased considerably from section H to section B due to the geometric modifications introduced. 5.3. Motion-dependent aerodynamic forces — aerodynamic derivatives Simulations of forced harmonic bending and twisting section motions are used for determination of motion-induced aerodynamic forces. Lift and moment time traces obtained from numerical simulations are processed to yield aerodynamic derivatives following the procedure presented by Larsen and Walther [10]. The G5 and G2 cross-sections investigated in Ref. [10] are almost identical to the H and B sections considered above, hence the simulated G5, G2 aerodynamic derivatives will be considered representative of the present H and B cross-sections and thus reviewed here. Fig. 8 superimposes the aerodynamic derivatives obtained from simulations on the wind tunnel data presented by Scanlan and Tomko [11]. Only the six major derivatives are reported in line with Scanlan and Tomko’s work. The remaining H* and A* derivatives are of little significance for practical flutter 4 4 predictions. A few comments are appropriate at this point. The simulated H*!H* and the 1 3 A* derivatives representative of the B section compare very well to the airfoil data (A). 1 The A* and A* display less correlation with the airfoil data, possibly due to leading 2 3 edge separation caused by the sharp-edged corners of the bridge sections. When comparing the simulations to the experimental bridge section data it is noticed that H* obtained from simulations is all together different. The A* and A* derivatives are, 2 2 3 however, in very good agreement with the derivatives measured for the bridge deck model. In case of the H section the A* derivative is the most important coefficient as 2 its change of sign (from negative at low reduced wind speeds to positive at high wind speeds) signifies one-degree-of-freedom torsional flutter. The simulations which are run at a forced twisting amplitude of 3° indicate a cross-over point for A* at about 2 twice the wind speed as compared to the wind tunnel data. The remaining aerodynamic derivatives for the H cross-section are in reasonable agreement with the experiments.
6. Influence of cross-section shape on wind response and stability The role of the section shape-dependent aerodynamic parameters on horizontal along-wind buffeting response, vertical vortex-induced response and critical wind speed for onset of flutter is illustrated through the set of expressions given in the appendix. Assuming similar structural properties and wind conditions for bridges involving the five cross-sections investigated allows assessment of their relative response to wind. Table 2 presents such an evaluation based on the C , CRMS and St D0 L coefficients summarised in Table 3 and the simulated aerodynamic derivatives given in Fig. 8. Cross-section H serves as reference and is assigned a unit response (1.0),
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Fig. 8. Comparison of simulated aerodynamic derivatives for the H and B cross-sections aerodynamic derivatives obtained from wind tunnel section model tests.
whereas the remaining section responses are evaluated relative to this. Critical wind speeds º for onset of flutter are calculated specifically in m/s using the structural data # of Fig. 2. It is noted from Table 4 that wind-induced response due to along-wind buffeting and vortex shedding is sensitive to the cross-section shape. The trapezoidal box section appears to be significantly less susceptible to wind response than the remaining
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A. Larsen/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 73–90 Table 4 Relative wind-induced response of bridges due to different cross-section shapes Cross-section designation
Horizontal buffeting response
Vertical vortex response
Critical wind speed º (m/s) #
H C R CE B
1.0 0.82 0.82 0.76 0.53
1.0 0.89 0.97 1.42 0.34
11.5 — — — 20.5
cross-sections. Also the flutter performance of trapezoidal box cross-section appears to be superior at least to the notoriously unstable H-shaped deck cross-section.
7. Case study — wind response of a 400 m main span cable-stayed bridge Numerical simulations using the discrete vortex method are run on a regular basis by the author’s company for assessment of the aerodynamic performance of new bridge projects or retrofits. The present case study considers a 400 m main span cable-stayed bridge, Fig. 9. The bridge was tendered with two alternative cross-sections: (A1) A composite cross-section composed of a concrete deck slab carried by rectangular box edge beams, plate cross-girders and longitudinals all in steel. (A2) A composite crosssection composed of a concrete deck slab supported by a closed steel box structure. Both alternatives were equipped with solid New Jersey type crash barriers which caused some concern with respect to the aerodynamic performance of the bridge. DVMFLOW simulations of steady-state wind load coefficients at !3°, 0° and 3° angle of attack were carried out for application to buffeting calculations and for identification of the lock-in wind speed for vortex-shedding excitation. Simulations of motion-induced aerodynamic loads were carried out to obtain aerodynamic derivatives for input to flutter routines. Vertical vortex-induced responses were simulated directly in DVMFLOW by allowing the cross-sections to be supported by vertical spring elements tuned to the lock-in frequency. Simulated flow fields about the alternative plate girder and box girder cross-sections are shown in Fig. 10. The flow about the plate girder cross-section forms large recirculating vortical structures below the deck in the compartments between the edge girders and the longitudinals. In contrast, the flow about the box girder cross-section is smooth along the slightly curved bottom plate. The differences in the respective flow fields carry over in the predicted aerodynamic properties and the bridge response as summarised in Table 5. For the present practical example it is noted that the box section A2 is aerodynamically superior to the plate section A1. The drag coefficient and the vertical
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Fig. 9. 400 m main span cable-stayed bridge studied by discrete vortex simulations.
Fig. 10. Simulated flow about alternative girder cross-sections for a 400 m cable-stayed bridge.
Table 5 Predicted aerodynamic drag C , Strouhal number St, vertical vortex-induced response h (in first vertical D0 bending mode), flutter mode and critical wind speed for onset of flutter º # Cross-section designation
Drag coefficient Strouhal no. C St D0
Plate section, A1 0.12 Box section, A2 0.07
0.13 0.16
Vertical vortex response h (m)
Flutter mode
Flutter wind speed º (m/s) #
0.055 0.034
1DOF 2DOF
130 210
vortex-induced response of the A2 cross-section is reduced by approximately 40% relative to the A1 cross-section, whereas the critical wind speed for onset of flutter is increased by 60%.
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8. Conclusion The present paper has discussed the use of two-dimensional numerical flow simulations in bridge aerodynamics. The paper has focused on the expected accuracy, correlation with wind tunnel test results and the application in bridge design studies for determination of cross-section shape on bridge response to wind. It is concluded that numerical simulations present a worthwhile alternative to section model testing in cases where the deck cross-section geometry investigated allow meaningful discretisation in two dimensions. From a bridge designers point of view numerical simulations appear as an efficient tool for weeding out inefficient cross-section alternatives before embarking on confirmatory wind tunnel testing.
Appendix A. Mathematical models of bridge response to wind A.1. Horizontal buffeting response to turbulent winds The buffeting theory developed by Davenport [3] considers each vibration mode receiving excitation by atmospheric turbulence as a one-degree-of-freedom (1DOF) oscillator. In this format root mean square bridge response p at the eigenfrequency x f of each individual horizontal mode of oscillation may be obtained as
A BA B S
oB3 p "C x D0 M*
º 21 fB 8
1 I p3(f #f ) u 4 !
S
fS ( f ) u p2 u
L
P 0
L e~C@s~s{@f@Uu(s) u(s@) ds ds@,
P 0
(A.1)
where f and f are the structural and aerodynamic damping levels relative to critical, 4 ! M*"m:Lu2(s) ds is the modal mass in the mode of motion considered, m is the 0 mass/unit length of structure, I is the (along-wind) turbulence intensity, fS ( f )/p2 is u u u the normalised power spectrum of turbulence, J:Le~C@s~{@f@Uu(s):Lu(s@) ds ds@ is the 0 0 spanwise joint acceptance function, C+4—8, and u(s), s and ¸ are mode shape spanwise coordinate and span length. The important thing to notice at this point is that the horizontal buffeting response is directly proportional to the drag coefficient C which again is a function of the D0 cross-section shape. The remaining parameters relate either to the structural properties of the bridge (modal mass and mode shape) or to the wind climatic conditions prevailing at the bridge site. A.2. Vertical vortex-induced response Vertical vortex-induced response of bridge structures may be treated in a modal format much the same way as the resonant horizontal buffeting response given above. Wyatt and Scruton [13] have proposed a 1DOF oscillator model in which the vertical
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87
periodic load due to vortex shedding excitation is represented by a root mean square lift coefficient CRMS to be obtained from wind tunnel section model tests. A slightly L modified version of the Wyatt/Scruton model yields the following root mean square vertical vortex shedding response p : h
A BA
CRMS p" L h St2
1 16p2(f #f ) 4 !
BA B
oBH2 :LDu(s)D ds 0 , :Lu2(s) ds m 0
(A.2)
where m is the cross-section mass/unit length and :LDu(s)D ds/:Lu2(s) ds is a factor '1 0 0 accounting for the effect of mode shape. It is noticed that the bridge response, at least to a first approximation, is proportional to the ratio of the root means square lift coefficient to the Strouhal number squared, items which again are functions of the cross-section shape. The remaining parameters relate to the structural properties of the bridge. A.3. Aerodynamic damping A measure for the aerodynamic damping f (relative to critical) is needed for ! carrying out response calculations for along-wind buffeting and cross-wind vortexshedding excitation. In the case of along-wind buffeting, the aerodynamic damping arises from a force opposing the motion in the direction of the mean wind. In this case, f is expressed in terms of the cross section drag coefficient: ! oºBC D0. f" ! 4pm
(A.3)
In the case of vertical vortex-shedding excitation, the aerodynamic damping arising from a cross-wind force opposing the vertical motion is negative. In applying the aerodynamic derivative formulation of motion-induced aerodynamic forces, Scanlan [12] the cross-wind aerodynamic damping is obtained as oB2H* 1. f "! ! 2m
(A.4)
A.4. Aeroelastic instability – flutter Two types of flutter instabilities are commonly encountered in bridge engineering: (1) 1DOF torsional flutter by which the girder responds to motion induced aerodynamic forces in a pure torsional mode. (2) 2DOF flutter by which the bridge girder responds in a combined bending and torsional mode due to cross-coupled motioninduced aerodynamic forces. Mathematical models for onset of one or two-degree-offreedom flutter instability are developed from similar modal concepts as the models for buffeting and vortex shedding response. The representation of the motion induced aerodynamic forces acting on a cross section is however slightly more complicated. A convenient framework for distinction of flutter type (one or two-degree-of-freedom)
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and prediction the critical wind speed for onset of flutter is given by Scanlan [12], who introduced a set of aerodynamic coefficients, the so-called aerodynamic derivatives, representing the motion induced aerodynamics of a given cross section. The aerodynamic derivatives which are found to be functions of reduced wind speed º/fB and cross section geometry may be related to the well known lift and moment coefficients in cases where the cross-section is undergoing forced harmonic bending or twisting motion [11]. A.5. One-degree-of-freedom torsional flutter A mathematical model for pure torsional flutter is presented by the following 1DOF oscillator assuming time complex harmonic twisting motion of the cross section (i is the imaginary unit): I[(u2!u2)#i2f u u]a"oº2B2 a 4 a
A B
uB 2 [i A*#A*]a, 3 2 º
(A.5)
where I is the cross-section mass moment of inertia/unit length, u and u are circular a eigenfrequency and circular frequency of motion and A* and A* are aerodynamic 3 2 derivatives representing aerodynamic damping and stiffness. The critical wind speed for onset of 1DOF torsional flutter is identified as the wind speed where the structural damping balances “negative” aerodynamic damping. From the equation of motion this condition is fulfilled for the following critical value of (A*) : 2# 2If (A*) " 4 2 # oB4
(A.6)
taking u +u. a If A* is plotted in a diagram as function of º/fB as is common practice, the critical 2 wind speed for onset of flutter is obtained as the abscissa (º/fB) to (A*) . Aerodynam# 2# ically speaking, 1DOF torsional flutter is distinguishable from 2DOF flutter by the fact that the A* aerodynamic derivative (which is proportional to the aerodynamic 2 damping in torsion) changes sign from negative at low º/fB to positive at some higher value of º/fB. A.6. Two-degree-of-freedom coupled flutter Cross-sections for which the A* aerodynamic derivative remains negative for all 2 reduced wind speeds º/fB (A* negative"positive aerodynamic damping in torsion) 2 are likely to display 2DOF coupled vertical/torsional flutter behaviour. This occurs at the wind speed where the motion-induced aerodynamic loads cause vertical and torsional frequencies of motion to collapse into one common frequency. A mathematical model for coupled vertical/torsional flutter is presented by the following set of one-degree-of-freedom oscillators assuming time complex harmonic vertical (h) and
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twisting (a) motion of the cross-section:
A BC
D
h uB 2 m[(u2!u2)#i2u uf]h"oº2B (iH*#H*) #(iH*#H*)a , 4 B 3 h h 1 2 º (A.7)
A BC
D
h uB 2 I[(u2!u2)#i2u uf]a"oº2B2 (iA*#A*) #(iA*#A*)a , 4 3 a a 1 2 B º (A.8) where u and u are the respective circular eigenfrequencies, u a common flutter h a frequency and H*2 , A*2 are the self-induced section loads — the aerodynamic 1 4 1 4 derivatives. It is noted that the aerodynamic loads introduces coupling between the equations for the vertical (h) and twisting (a) motion. Introducing the frequency ratio c"u /u a h and the frequency ratio X"u/u and rearranging the equations of motion yields the h flutter determinant which, when set equal to zero defines the flutter point:
K
A
oB2 oB2 1!X2! H*X2#i 2fX! H*X2 4 1 m m oB4 oB4 A*X2!i A*X2 ! 4 1 I I
B
oB2 oB2 ! H*X2! H*X2 3 2 m m
A
B
oB4 oB4 c2!X2! A*X2#i 2cXf! A*X2 3 2 I I
K
"0.
The flutter determinant defines a fourth-order real and a third-order imaginary algebraic equation to be solved for X introducing the H*2 , A*2 coefficients 1 4 1 4 obtained at successive values of the reduced wind speed º/fB. Onset of 2DOF flutter will occur at the particular reduced wind speed (º/fB) where the roots of the real and # imaginary equations X3%!-, X*.!' are identical"X . Finally, the critical wind speed # º for onset of 2DOF coupled flutter is obtained as # º º" f BX . (A.9) # fB ) # # For a more detailed and complete description of the use of aerodynamic crosssection data in bridge response analyses the reader is referred to the state-of-the-artreview by Scanlan [12].
A B
References [1] F.B. Farquharson, Aerodynamic stability of suspension bridges, University of Washington Experimental Station, Bull. 116, Part I—V, 1949—54. [2] R.A. Frazer, C. Scruton, A summarised account of the severn bridge aerodynamic investigation, NPL Aero Report, 222, London, HMSO, 1952. [3] A.G. Davenport, Buffeting of a suspension bridge by storm winds, J. Struct. Div. ASCE (1962) 233—264. [4] A. Larsen, Aerodynamic aspects of the final design of the 1624 m suspension bridge across the great belt, J. Wind Eng. Ind. Aerodyn. 48 (1993) 261—285.
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[5] S. Kuroda, Numerical simulation of flow around bridge, Reprint IHI Eng. Rev. 29 (2) (1996). [6] J.H. Walther, Discrete vortex method for two-dimensional flow past bodies of arbitrary shape undergoing prescribed rotary and translatory motion, AFM-94-11, Ph.D. Thesis, Dept. of Fluid Mechanics, Technical University of Denmark, 1994. [7] P. Selvam, M.J. Tarini, A. Larsen, Computer modelling of flow around bridges using LES and FEM, Paper presented at 8th US National Conf. on Wind Eng. Johns Hopkins University, 1997. [8] A. Larsen, J.H. Walther, Aeroelastic analysis of bridge girder sections based on discrete vortex simulations, Paper Presented at 2nd Int. Conf. on Comput. Wind Eng., Colorado State University, 1996. [9] J.H. Walther, A. Larsen, 2D Discrete vortex method for application to bluff body aerodynamics, 1996. [10] A. Larsen, J.H Walther, Discrete vortex simulation of flow around five generic bridge deck sections, Paper Presented at 8th US National Conf. Wind Eng. Johns Hopkins University, 1997. [11] R.H. Scanlan, J.J. Tomko, Airfoil and bridge deck flutter derivatives. J. Mech. Div., EM6, ASCE, 1971. [12] R.H. Scanlan, State-of-the-art methods for calculating flutter, vortex-induced and buffeting response of bridge structures, Federal Highway Administration, Report No. FHWA / RD-80 / 050. Washington, DC, 1981. [13] T.A. Wyatt, C. Scruton, A brief survey of the aerodynamic stability problems of bridges, In: Bridge Aerodynamics, Institution of Civil Engineers, London, 1981.
Advances in aeroelastic analyses of suspension and cable-stayed bridges Allan Larsen* COWI Consulting Engineers and Planners AS, Parallelvej 15, DK-2800 Lyngby, Denmark
Abstract Cross-section shape is an important parameter for the wind response and aeroelastic stability of long span suspension and cable-stayed bridges. Numerical simulation methods have now been developed to a stage where assessment of the effect of practical cross-section shapes on bridge response is possible. The present paper reviews selected numerical simulations carried out for a long-span suspension bridge using finite difference and discrete vortex methods. Comparison of simulations to existing wind tunnel data is discussed. Further, the paper addresses the aerodynamics and structural response of four generic cross-section shapes developed from the well-known plate girder section of the first Tacoma Narrows Bridge. Finally a case study involving the wind response of a 400 m main span cable-stayed bridge is discussed. ( 1998 Elsevier Science Ltd. All rights reserved. Keywords: Numerical methods; Bridge aerodynamics; Buffeting; Vortex shedding excitation; Aeroelastic instability
1. Introduction The engineering discipline of bridge-aerodynamics was born of the spray rising from the fall of the first Tacoma Narrows Bridge into the Puget Sound in 1940. In his monumental investigation of the bridge collapse, Farquharson [1] covered a wide range of technical aspects ranging from experimental techniques over aerodynamics and structural dynamics to guidelines for bridge design. Probably, the singlemost important finding of the Tacoma Narrows investigation was that vortex-shedding excitation and flutter instability of a complete suspension bridge could be accurately represented by a spring-supported section model of the deck structure. This important
* E-mail: [email protected]. 0167-6105/98/$19.00 ( 1998 Elsevier Science Ltd. All rights reserved. PII: S 0 1 6 7 - 6 1 0 5 ( 9 8 ) 0 0 0 0 7 - 5
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result was later corroborated by wind tunnel investigations conducted in the United Kingdom in connection with design of a suspension bridge for crossing of the river Severn, Frazer and Scruton [2]. Further versatility was added to section model testing by Davenport [3], who advanced a method for assessment of buffeting response of complete bridge structures to turbulent wind based on aerodynamic section data. Needless to say, section model testing offers substantial savings relative to testing of full aeroelastic bridge section models. Structural analyses of bridges have moved from laboratory testing of physical models to computer-based finite element modelling. This development has allowed the designer to experiment with different structural systems and configurations without resorting to expensive and time-consuming physical testing. Aerodynamic analysis of bridges has not seen a similar development due to the complexity of the fluid dynamic phenomena involved, hence most aerodynamic analyses of bridge structures are still restricted by aerodynamic data obtained from wind tunnel testing. Numerical fluid dynamic models and computer capacity have developed over the past decade to a stage where the bridge designer may start to exploit these new techniques in actual design work in much the same way as physical section model tests. In particular, numerical simulations appear well suited for design studies of the effect of cross-section shape on bridge response to wind loading, thus presenting an efficient tool for weeding out inefficient cross-sections before embarking on confirmatory wind tunnel testing. The present paper will highlight some recent comparisons between wind tunnel section model results and numerical simulations. The main body of the paper will be devoted to a design study of the effect of cross-section shape on bridge response taking the first Tacoma Narrows Bridge as an example. Finally the paper will outline a numerical design study carried out for determination of the most favourable cross-section shape for a 400 m main span cable-stayed bridge.
2. Model tests and numerical simulations The wind design of the East Bridge was carried out in the time span 1989—1992 and was based on extensive wind tunnel section model testing. Since that time numerical methods have developed to the extent that two-dimensional aerodynamic section data may be calculated with acceptable accuracy for design studies. In order to illustrate this point, measured and calculated wind load coefficients for the girder cross-sections of the East Bridge suspended spans and approach spans, shown in Fig. 1, will be compared. 2.1. East Bridge suspended spans Steady-state wind loads for the cross-section of the East Bridge suspended spans have been reported by two different workers using different numerical simulation techniques. Kuroda [5] applied a gird-based finite-difference method (FDM) using the pseudo-compressibility technique for solution of the incompressible two-dimensional Navier—Stokes equations at Reynolds number Re"3]105 (based
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75
Fig. 1. Girder cross-sections of the East Bridge suspended spans and approach spans. Table 1 Comparison of steady-state wind load coefficients obtained from numerical simulations and wind tunnel testing of a 1 : 80 section model. East Bridge Suspended spans Method
C
FDM [5] DVM [6] Experiment
0.071 0.061 0.081
D0
(-) C
L0
(-)
!0.100 0.000 0.067
C
M0
(-)
0.025 0.027 0.028
dC /da (-/rad) L
dC /da (-/rad) M
6.48 4.13 4.37
1.15 1.15 1.17
on cross-section width B). Walther [6] applied the grid-free discrete vortex method (DVM) for solution of the two-dimensional vorticity equation representing the flow around the cross-section at Re"105. The wind loads reported are made nondimensional through division with the dynamic head 1oº2 and section width B: 2 ¸ M D , C " , C " . (1) C " L M 1oº2B2 D 1oº2B 1oº2B 2 2 2 Table 1 compares simulated wind load coefficients C , C , C at zero angle of D0 L0 M0 attack and lift and moment slopes dC /da, dC /da to experimental values obtained L M from wind tunnel testing of a 1 : 80 section model as reported by Larsen [4]. Satisfactory agreement between simulations and experiment is demonstrated for most of the coefficients with the exception of dC /da obtained from the FDM L simulations. This coefficient is 48% in excess of the experimental data. In comparing C values it shall be remembered that the physical section model was equipped with D0 light tubular railings and crash barriers, whereas the numerical geometry models only reproduced the gross trapezoidal cross-section shape. Simple calculations allowing each of the railing components to be exposed to the free stream wind speed yield a drag contribution of *C "0.023 which brings simulations and experiment in D0 better agreement. 2.2. East Bridge approach spans Similar to the cross-section of the suspended spans the approach spans (Fig. 2, right) have been subject to two-dimensional numerical flow simulations. Selvam
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A. Larsen/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 73–90 Table 2 Comparison of drag coefficient and Strouhal number obtained from numerical simulations and wind tunnel testing of a 1 : 80 scale section model. East Bridge Approach spans Method
C
FDM — LES [7] DVM [8] Experiment
0.187 0.179 0.190
D0
(-)
St (-) 0.166—0.199 0.167 0.170
et al. [7] applied a finite-difference method including a large eddy simulation (LES) model for prediction of the important drag loading and Strouhal number and vortex-shedding frequency. Larsen and Walther [8] report similar results obtained by means of the discrete vortex method. Table 2 offers a comparison between numerical simulations and experimental results of drag coefficient and Strouhal number: D fH C " , St" , D 1oº2B º 2
(2)
where H is cross-wind section depth (H"7.0 m) and f is vortex shedding frequency. As in the case of the suspended span cross-section the numerical simulations are in fair agreement with experimental data. Again railings and crash barriers were not included in the numerical models, hence slightly lower C values are to be expected D0 when comparing with the experiment. Further comparisons between experiment and numerical simulations are presented by Larsen and Walther [8] for the H-shaped cross-section of the first Tacoma Narrows Bridge and for a twin-box cross-section developed for a fixed link across the Straits of Gibraltar. These results are equally promising indicating that numerical simulations of flow around bridge girders are worth while in bridge design and retrofit studies.
3. Models for bridge response to wind Bridge response to wind is mainly governed by the aerodynamic properties of the girder cross-sections, structural parameters such as mass, mass moment of inertia, eigenfrequencies and damping and for buffeting response the turbulence properties of the wind field. The appendix offers a brief run down of mathematical models which may be used for a first-order assessment of the three most important types of wind-induced response: (1) along-wind buffeting response (drag direction), (2) vertical
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77
vortex-shedding excitation and (3) critical wind speed for onset of flutter. Besides being of practical use, the models illustrate that bridge buffeting, vortex-shedding excitation and flutter stability may be calculated, once drag coefficient C , root mean D0 square lift coefficient CRMS, Strouhal number St and aerodynamic derivatives H* , L 1..4 A* are available for a given bridge girder cross-section. A more complete descrip1..4 tion of the use of aerodynamic cross-section data in bridge response analyses is offered by Scanlan [12]. The following section will focus on the above-mentioned aerodynamic properties and simulation by the discrete vortex method.
4. Discrete vortex method for 2D bridge deck cross-sections A distinct feature of flow about bluff bodies, stationary or in time-dependent motion, is the shedding of vorticity in the wake which balances the change of fluid momentum along the body surface. The vorticity shed at an instant in time is convected downstream by the mean wind speed but continues to affect the aerodynamic loads on the body. A mathematical model for the flow around bluff bodies was developed within the framework of the discrete vortex method and programmed for computer by Walther [6]. The resulting numerical code DVMFLOW establishes a “grid-free” time-marching simulation of the vorticity equation well suited for simulation of 2D bluff body flows. An outline of the mathematical model and the simulated flow about a flat plate is presented by Walther and Larsen [9]. The input to DVMFLOW simulations is a boundary panel model of the cross-section contour. The output of DVMFLOW simulations is time progressions of surface pressures and section loads (drag, lift and moment). In addition, maps of the flow field (vector plots), vortex positions and streamlines at prescribed time steps are available. Steady-state wind load coefficients and Strouhal number are obtained from time averages and frequency analysis of simulated loads on stationary panel models. Aerodynamic derivatives are obtained from post-processing of simulated time series of forced harmonic motion as detailed by Larsen [10] in a numerical investigation of five generic bridge deck cross-sections tested in a wind tunnel by Scanlan and Tomko [11].
5. Five Bridge deck cross-sections — an example The lively wind response (the galloping) and the final collapse of the first Tacoma Narrows Bridge, Fig. 2, was established to be due to the aerodynamically unfavorable cross-section shape and lightness of the bridge structure. Although a number of investigations has pointed out that H-shaped cross-sections similar to the first Tacoma Narrows are undesirable from an aerodynamic point of view they remain attractive from an economic point of view as well as due to ease of fabrication. The present example will thus consider the aerodynamic effect of four simple modifications of the parent H-shaped cross-section.
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Fig. 2. First Tacoma Narrows Bridge. Girder deck cross-section, elevation and structural data applicable to the first asymmetric mode of vibration [1].
Fig. 3. Cross-section shapes considered in the present study.
5.1. Cross-sections investigated The five cross-sections investigated are shown in Fig. 3. The parent cross-section denoted H is a slightly simplified version of the first Tacoma Narrows deck omitting cross-girders and curbs but reproducing the longitudinal edge girders and floor slab. Section C (channel type) is obtained from the H section simply by adding a top plate. Section R (rectangular type) is obtained by adding a bottom-plate to the C section. The CE section (channel/edge) is obtained
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from the C section by adding triangular edge-fairings to the C section. Finally the B section (box type) is obtained by adding the triangular edge-fairings to the R section or closing the bottom of the CE section. All dimensions in Fig. 3 are referred to the width B of the parent H section. The circumference of each cross-section was subdivided into a total of 300 surface vortex panels. The discretisation allowed flow at Reynolds number Re"105 (Re"ºB/l) to be simulated. 5.2. Simulation of flow about stationary sections Drag coefficient C , root mean square lift coefficient CRMS and Strouhal number St L D0 are obtained from simulations of the flow about the five cross-sections fixed in space. An angle of attack of 0° of the wind flow was assumed (angle between flow direction and section chord). Each simulation was run for 30 non-dimensional time units ¹"tº/B where t is the time, º is the flow speed and B is the chord length. A non-dimensional time increment *¹"0.025 was adopted throughout the simulations. At each time step the cross-section surface pressure distribution was computed from the local flux of surface velocity. The section surface pressures were finally integrated along the contour to form time traces of the section drag D and lift ¸ forces. Lastly, the computed aerodynamic forces were expressed in non-dimensional form following Eq. (1). Fig. 4 shows an example of the simulated time traces of C and D0 C obtained for cross-section H. L The C trace displays initial very high values associated with the instantaneous D start up of the flow simulation. After an exponential decay the C trace settles around D a mean value C "0.28 after approximately 5 non-dimensional time units. D0 C "0.28 is in satisfactory agreement with C "0.29!0.30 reported by FarD0 D0 quharson [1]. The C trace develops very distinct oscillations with period ¹+1.7 L associated with formation of vortex roll up in the wake — the well-known von Ka´rma´n vortex street, Fig. 5. The wake pattern obtained from simulation of the flow
Fig. 4. Simulated time traces of drag coefficient C and lift coefficient C for cross-section H. D L
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Fig. 5. Formation of von Ka´rma´n vortex street in the wake of cross-section H.
Fig. 6. Formation of von Ka´rma´n vortex street in the wake of cross-section B.
Fig. 7. Simulated time traces of drag coefficient C and lift coefficient C for cross-section B. D L
around cross-section B is shown in Fig. 6, whereas simulated time traces of C and D C are shown in Fig. 7. A summary of aerodynamic data for all sections is given in L Table 1. From Table 3 it is noted that the closed-box section B displays better aerodynamic performance than the remaining cross-sections, i.e. lower C and CRMS. The parent D L cross-section H appears to yield the worst aerodynamic performance.
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A. Larsen/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 73–90 Table 3 Flow fields in the vicinity of the cross sections and C , CRMS and St values extracted D0 L Cross section geometry and flow patterns
C D0
CRMS L
St
0.28
0.37
0.11
0.23
0.33
0.11
0.23
0.24
0.09
0.16
0.34
0.09
0.11
0.17
0.13
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A. Larsen/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 73–90
By comparing Figs. 4 and 7 it is noted that the drag loading and the oscillating lift have decreased considerably from section H to section B due to the geometric modifications introduced. 5.3. Motion-dependent aerodynamic forces — aerodynamic derivatives Simulations of forced harmonic bending and twisting section motions are used for determination of motion-induced aerodynamic forces. Lift and moment time traces obtained from numerical simulations are processed to yield aerodynamic derivatives following the procedure presented by Larsen and Walther [10]. The G5 and G2 cross-sections investigated in Ref. [10] are almost identical to the H and B sections considered above, hence the simulated G5, G2 aerodynamic derivatives will be considered representative of the present H and B cross-sections and thus reviewed here. Fig. 8 superimposes the aerodynamic derivatives obtained from simulations on the wind tunnel data presented by Scanlan and Tomko [11]. Only the six major derivatives are reported in line with Scanlan and Tomko’s work. The remaining H* and A* derivatives are of little significance for practical flutter 4 4 predictions. A few comments are appropriate at this point. The simulated H*!H* and the 1 3 A* derivatives representative of the B section compare very well to the airfoil data (A). 1 The A* and A* display less correlation with the airfoil data, possibly due to leading 2 3 edge separation caused by the sharp-edged corners of the bridge sections. When comparing the simulations to the experimental bridge section data it is noticed that H* obtained from simulations is all together different. The A* and A* derivatives are, 2 2 3 however, in very good agreement with the derivatives measured for the bridge deck model. In case of the H section the A* derivative is the most important coefficient as 2 its change of sign (from negative at low reduced wind speeds to positive at high wind speeds) signifies one-degree-of-freedom torsional flutter. The simulations which are run at a forced twisting amplitude of 3° indicate a cross-over point for A* at about 2 twice the wind speed as compared to the wind tunnel data. The remaining aerodynamic derivatives for the H cross-section are in reasonable agreement with the experiments.
6. Influence of cross-section shape on wind response and stability The role of the section shape-dependent aerodynamic parameters on horizontal along-wind buffeting response, vertical vortex-induced response and critical wind speed for onset of flutter is illustrated through the set of expressions given in the appendix. Assuming similar structural properties and wind conditions for bridges involving the five cross-sections investigated allows assessment of their relative response to wind. Table 2 presents such an evaluation based on the C , CRMS and St D0 L coefficients summarised in Table 3 and the simulated aerodynamic derivatives given in Fig. 8. Cross-section H serves as reference and is assigned a unit response (1.0),
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83
Fig. 8. Comparison of simulated aerodynamic derivatives for the H and B cross-sections aerodynamic derivatives obtained from wind tunnel section model tests.
whereas the remaining section responses are evaluated relative to this. Critical wind speeds º for onset of flutter are calculated specifically in m/s using the structural data # of Fig. 2. It is noted from Table 4 that wind-induced response due to along-wind buffeting and vortex shedding is sensitive to the cross-section shape. The trapezoidal box section appears to be significantly less susceptible to wind response than the remaining
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A. Larsen/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 73–90 Table 4 Relative wind-induced response of bridges due to different cross-section shapes Cross-section designation
Horizontal buffeting response
Vertical vortex response
Critical wind speed º (m/s) #
H C R CE B
1.0 0.82 0.82 0.76 0.53
1.0 0.89 0.97 1.42 0.34
11.5 — — — 20.5
cross-sections. Also the flutter performance of trapezoidal box cross-section appears to be superior at least to the notoriously unstable H-shaped deck cross-section.
7. Case study — wind response of a 400 m main span cable-stayed bridge Numerical simulations using the discrete vortex method are run on a regular basis by the author’s company for assessment of the aerodynamic performance of new bridge projects or retrofits. The present case study considers a 400 m main span cable-stayed bridge, Fig. 9. The bridge was tendered with two alternative cross-sections: (A1) A composite cross-section composed of a concrete deck slab carried by rectangular box edge beams, plate cross-girders and longitudinals all in steel. (A2) A composite crosssection composed of a concrete deck slab supported by a closed steel box structure. Both alternatives were equipped with solid New Jersey type crash barriers which caused some concern with respect to the aerodynamic performance of the bridge. DVMFLOW simulations of steady-state wind load coefficients at !3°, 0° and 3° angle of attack were carried out for application to buffeting calculations and for identification of the lock-in wind speed for vortex-shedding excitation. Simulations of motion-induced aerodynamic loads were carried out to obtain aerodynamic derivatives for input to flutter routines. Vertical vortex-induced responses were simulated directly in DVMFLOW by allowing the cross-sections to be supported by vertical spring elements tuned to the lock-in frequency. Simulated flow fields about the alternative plate girder and box girder cross-sections are shown in Fig. 10. The flow about the plate girder cross-section forms large recirculating vortical structures below the deck in the compartments between the edge girders and the longitudinals. In contrast, the flow about the box girder cross-section is smooth along the slightly curved bottom plate. The differences in the respective flow fields carry over in the predicted aerodynamic properties and the bridge response as summarised in Table 5. For the present practical example it is noted that the box section A2 is aerodynamically superior to the plate section A1. The drag coefficient and the vertical
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Fig. 9. 400 m main span cable-stayed bridge studied by discrete vortex simulations.
Fig. 10. Simulated flow about alternative girder cross-sections for a 400 m cable-stayed bridge.
Table 5 Predicted aerodynamic drag C , Strouhal number St, vertical vortex-induced response h (in first vertical D0 bending mode), flutter mode and critical wind speed for onset of flutter º # Cross-section designation
Drag coefficient Strouhal no. C St D0
Plate section, A1 0.12 Box section, A2 0.07
0.13 0.16
Vertical vortex response h (m)
Flutter mode
Flutter wind speed º (m/s) #
0.055 0.034
1DOF 2DOF
130 210
vortex-induced response of the A2 cross-section is reduced by approximately 40% relative to the A1 cross-section, whereas the critical wind speed for onset of flutter is increased by 60%.
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8. Conclusion The present paper has discussed the use of two-dimensional numerical flow simulations in bridge aerodynamics. The paper has focused on the expected accuracy, correlation with wind tunnel test results and the application in bridge design studies for determination of cross-section shape on bridge response to wind. It is concluded that numerical simulations present a worthwhile alternative to section model testing in cases where the deck cross-section geometry investigated allow meaningful discretisation in two dimensions. From a bridge designers point of view numerical simulations appear as an efficient tool for weeding out inefficient cross-section alternatives before embarking on confirmatory wind tunnel testing.
Appendix A. Mathematical models of bridge response to wind A.1. Horizontal buffeting response to turbulent winds The buffeting theory developed by Davenport [3] considers each vibration mode receiving excitation by atmospheric turbulence as a one-degree-of-freedom (1DOF) oscillator. In this format root mean square bridge response p at the eigenfrequency x f of each individual horizontal mode of oscillation may be obtained as
A BA B S
oB3 p "C x D0 M*
º 21 fB 8
1 I p3(f #f ) u 4 !
S
fS ( f ) u p2 u
L
P 0
L e~C@s~s{@f@Uu(s) u(s@) ds ds@,
P 0
(A.1)
where f and f are the structural and aerodynamic damping levels relative to critical, 4 ! M*"m:Lu2(s) ds is the modal mass in the mode of motion considered, m is the 0 mass/unit length of structure, I is the (along-wind) turbulence intensity, fS ( f )/p2 is u u u the normalised power spectrum of turbulence, J:Le~C@s~{@f@Uu(s):Lu(s@) ds ds@ is the 0 0 spanwise joint acceptance function, C+4—8, and u(s), s and ¸ are mode shape spanwise coordinate and span length. The important thing to notice at this point is that the horizontal buffeting response is directly proportional to the drag coefficient C which again is a function of the D0 cross-section shape. The remaining parameters relate either to the structural properties of the bridge (modal mass and mode shape) or to the wind climatic conditions prevailing at the bridge site. A.2. Vertical vortex-induced response Vertical vortex-induced response of bridge structures may be treated in a modal format much the same way as the resonant horizontal buffeting response given above. Wyatt and Scruton [13] have proposed a 1DOF oscillator model in which the vertical
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87
periodic load due to vortex shedding excitation is represented by a root mean square lift coefficient CRMS to be obtained from wind tunnel section model tests. A slightly L modified version of the Wyatt/Scruton model yields the following root mean square vertical vortex shedding response p : h
A BA
CRMS p" L h St2
1 16p2(f #f ) 4 !
BA B
oBH2 :LDu(s)D ds 0 , :Lu2(s) ds m 0
(A.2)
where m is the cross-section mass/unit length and :LDu(s)D ds/:Lu2(s) ds is a factor '1 0 0 accounting for the effect of mode shape. It is noticed that the bridge response, at least to a first approximation, is proportional to the ratio of the root means square lift coefficient to the Strouhal number squared, items which again are functions of the cross-section shape. The remaining parameters relate to the structural properties of the bridge. A.3. Aerodynamic damping A measure for the aerodynamic damping f (relative to critical) is needed for ! carrying out response calculations for along-wind buffeting and cross-wind vortexshedding excitation. In the case of along-wind buffeting, the aerodynamic damping arises from a force opposing the motion in the direction of the mean wind. In this case, f is expressed in terms of the cross section drag coefficient: ! oºBC D0. f" ! 4pm
(A.3)
In the case of vertical vortex-shedding excitation, the aerodynamic damping arising from a cross-wind force opposing the vertical motion is negative. In applying the aerodynamic derivative formulation of motion-induced aerodynamic forces, Scanlan [12] the cross-wind aerodynamic damping is obtained as oB2H* 1. f "! ! 2m
(A.4)
A.4. Aeroelastic instability – flutter Two types of flutter instabilities are commonly encountered in bridge engineering: (1) 1DOF torsional flutter by which the girder responds to motion induced aerodynamic forces in a pure torsional mode. (2) 2DOF flutter by which the bridge girder responds in a combined bending and torsional mode due to cross-coupled motioninduced aerodynamic forces. Mathematical models for onset of one or two-degree-offreedom flutter instability are developed from similar modal concepts as the models for buffeting and vortex shedding response. The representation of the motion induced aerodynamic forces acting on a cross section is however slightly more complicated. A convenient framework for distinction of flutter type (one or two-degree-of-freedom)
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A. Larsen/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 73–90
and prediction the critical wind speed for onset of flutter is given by Scanlan [12], who introduced a set of aerodynamic coefficients, the so-called aerodynamic derivatives, representing the motion induced aerodynamics of a given cross section. The aerodynamic derivatives which are found to be functions of reduced wind speed º/fB and cross section geometry may be related to the well known lift and moment coefficients in cases where the cross-section is undergoing forced harmonic bending or twisting motion [11]. A.5. One-degree-of-freedom torsional flutter A mathematical model for pure torsional flutter is presented by the following 1DOF oscillator assuming time complex harmonic twisting motion of the cross section (i is the imaginary unit): I[(u2!u2)#i2f u u]a"oº2B2 a 4 a
A B
uB 2 [i A*#A*]a, 3 2 º
(A.5)
where I is the cross-section mass moment of inertia/unit length, u and u are circular a eigenfrequency and circular frequency of motion and A* and A* are aerodynamic 3 2 derivatives representing aerodynamic damping and stiffness. The critical wind speed for onset of 1DOF torsional flutter is identified as the wind speed where the structural damping balances “negative” aerodynamic damping. From the equation of motion this condition is fulfilled for the following critical value of (A*) : 2# 2If (A*) " 4 2 # oB4
(A.6)
taking u +u. a If A* is plotted in a diagram as function of º/fB as is common practice, the critical 2 wind speed for onset of flutter is obtained as the abscissa (º/fB) to (A*) . Aerodynam# 2# ically speaking, 1DOF torsional flutter is distinguishable from 2DOF flutter by the fact that the A* aerodynamic derivative (which is proportional to the aerodynamic 2 damping in torsion) changes sign from negative at low º/fB to positive at some higher value of º/fB. A.6. Two-degree-of-freedom coupled flutter Cross-sections for which the A* aerodynamic derivative remains negative for all 2 reduced wind speeds º/fB (A* negative"positive aerodynamic damping in torsion) 2 are likely to display 2DOF coupled vertical/torsional flutter behaviour. This occurs at the wind speed where the motion-induced aerodynamic loads cause vertical and torsional frequencies of motion to collapse into one common frequency. A mathematical model for coupled vertical/torsional flutter is presented by the following set of one-degree-of-freedom oscillators assuming time complex harmonic vertical (h) and
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twisting (a) motion of the cross-section:
A BC
D
h uB 2 m[(u2!u2)#i2u uf]h"oº2B (iH*#H*) #(iH*#H*)a , 4 B 3 h h 1 2 º (A.7)
A BC
D
h uB 2 I[(u2!u2)#i2u uf]a"oº2B2 (iA*#A*) #(iA*#A*)a , 4 3 a a 1 2 B º (A.8) where u and u are the respective circular eigenfrequencies, u a common flutter h a frequency and H*2 , A*2 are the self-induced section loads — the aerodynamic 1 4 1 4 derivatives. It is noted that the aerodynamic loads introduces coupling between the equations for the vertical (h) and twisting (a) motion. Introducing the frequency ratio c"u /u a h and the frequency ratio X"u/u and rearranging the equations of motion yields the h flutter determinant which, when set equal to zero defines the flutter point:
K
A
oB2 oB2 1!X2! H*X2#i 2fX! H*X2 4 1 m m oB4 oB4 A*X2!i A*X2 ! 4 1 I I
B
oB2 oB2 ! H*X2! H*X2 3 2 m m
A
B
oB4 oB4 c2!X2! A*X2#i 2cXf! A*X2 3 2 I I
K
"0.
The flutter determinant defines a fourth-order real and a third-order imaginary algebraic equation to be solved for X introducing the H*2 , A*2 coefficients 1 4 1 4 obtained at successive values of the reduced wind speed º/fB. Onset of 2DOF flutter will occur at the particular reduced wind speed (º/fB) where the roots of the real and # imaginary equations X3%!-, X*.!' are identical"X . Finally, the critical wind speed # º for onset of 2DOF coupled flutter is obtained as # º º" f BX . (A.9) # fB ) # # For a more detailed and complete description of the use of aerodynamic crosssection data in bridge response analyses the reader is referred to the state-of-the-artreview by Scanlan [12].
A B
References [1] F.B. Farquharson, Aerodynamic stability of suspension bridges, University of Washington Experimental Station, Bull. 116, Part I—V, 1949—54. [2] R.A. Frazer, C. Scruton, A summarised account of the severn bridge aerodynamic investigation, NPL Aero Report, 222, London, HMSO, 1952. [3] A.G. Davenport, Buffeting of a suspension bridge by storm winds, J. Struct. Div. ASCE (1962) 233—264. [4] A. Larsen, Aerodynamic aspects of the final design of the 1624 m suspension bridge across the great belt, J. Wind Eng. Ind. Aerodyn. 48 (1993) 261—285.
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A. Larsen/J. Wind Eng. Ind. Aerodyn. 74–76 (1998) 73–90
[5] S. Kuroda, Numerical simulation of flow around bridge, Reprint IHI Eng. Rev. 29 (2) (1996). [6] J.H. Walther, Discrete vortex method for two-dimensional flow past bodies of arbitrary shape undergoing prescribed rotary and translatory motion, AFM-94-11, Ph.D. Thesis, Dept. of Fluid Mechanics, Technical University of Denmark, 1994. [7] P. Selvam, M.J. Tarini, A. Larsen, Computer modelling of flow around bridges using LES and FEM, Paper presented at 8th US National Conf. on Wind Eng. Johns Hopkins University, 1997. [8] A. Larsen, J.H. Walther, Aeroelastic analysis of bridge girder sections based on discrete vortex simulations, Paper Presented at 2nd Int. Conf. on Comput. Wind Eng., Colorado State University, 1996. [9] J.H. Walther, A. Larsen, 2D Discrete vortex method for application to bluff body aerodynamics, 1996. [10] A. Larsen, J.H Walther, Discrete vortex simulation of flow around five generic bridge deck sections, Paper Presented at 8th US National Conf. Wind Eng. Johns Hopkins University, 1997. [11] R.H. Scanlan, J.J. Tomko, Airfoil and bridge deck flutter derivatives. J. Mech. Div., EM6, ASCE, 1971. [12] R.H. Scanlan, State-of-the-art methods for calculating flutter, vortex-induced and buffeting response of bridge structures, Federal Highway Administration, Report No. FHWA / RD-80 / 050. Washington, DC, 1981. [13] T.A. Wyatt, C. Scruton, A brief survey of the aerodynamic stability problems of bridges, In: Bridge Aerodynamics, Institution of Civil Engineers, London, 1981.