prediction of aeroelastic stability of suspension bridges during erection

JOURNAL OF

Mmlm Wmm,W Journal of Wind Engineering ~ ( ~ 1 ~

and Industrial Aerodynamics 72 (1997) 265-274 ELSEVIER

Prediction of aeroelastic stability of suspension bridges during erection

Allan Larsen

COW1 Consulting Engineers and Planners, 15 Parallelvej. DK-2800 Lynglv, Denmark

Abstract

During erection of long-span suspension bridges the stiffening girder lacks torsional continu- ity. Thus, it becomes important to consider flutter-type aeroelastic instabilities. During the early stages of deck erection, the critical wind speed for onset of flutter becomes particularly low and in this case it may prove interesting to consider flutter control by provision of eccentric ballast. The present paper extends standard bridge flutter routines based on measured aerodynamic derivatives to include the effect of eccentric deployment of ballast. The potential enhancement of the critical wind speed by this method is discussed and the importance of applying actual measured aerodynamic derivatives for the bridge section in question is emphasised as opposed to the use of theoretical "flat plate" aerodynamic derivatives.

I. Introduction

Contemporary suspension bridges are often built with shallow "streamlined" box girders which combine a high torsional stiffness and a high torsional/vertical frequency ratio (typically 2.5-3.0) with an aerodynamically favourable shape. For spans in the range 1500-2000m these design measures are usually sufficient to ensure a critical wind speed for onset of flutter well above the design wind speed adopted for the bridge site. Aeroelastic stability problems may, however, arise during deck erection, mainly due to temporary lack of torsional stiffness and due to low frequency ratios (1.0-2.0) in combination with full exposure of the girder surface to the wind. Potential stability problems can be avoided by restricting deck erection to periods with calm weather or by provision of temporary means for enhancement of the critical wind speed.

2. Enhancement of aeroelastic stability during erection

Brancaleoni [1] has discussed methods for temporary enhancement of the critical wind speed of suspension bridges during erection. Two methods, augmentation of the

0167-6105/97/$17.00 ~: 1997 Elsevier Science B.V. All rights reserved. PII S01 67-61 05 (97)00248- 1

266 .1. LaJ:s~l ./ tlSml l:'n.'.,', hid tcro,/l'n 72 1"1997) 265 274

structural damping and provision of ballast at the windward cable plane were investigated in connection with construction of the Humber Bridge in England. Augmentation of the structural damping was deemed impractical, but eccentric ballasting was found to bc practical and was successfully employed by the superstruc- ture contractor. Ballasting was accomplished through deployment of large-size water-filled plastic foil containers on the cantilevered footways just outboard of the cable planes as illustrated in Fig. I. In the evcnl of high winds, water could be drained from the leeward containers thus causing a windward shift of the deck centre of gravity.

The physical principle inw~lvcd in flutter control by eccentric ballasting is to reduce the aerodynamic moment acting on the centre of rotation of the deck as illustrated in Fig. 2. In case of the unloaded section, the aerodynamic moment is obtained by multiplying the aerodynamic lift with the distance (lever L = B / 4 )

from the aerodynamic centre A(' to the centre of gravity CG, (centre of rotation) situated in the deck ccntrclinc. If additional mass is introduced at the windward edgc of the deck, the CG will shift towards AC. This action will reduce the lever IL < B /4 ) and thus the aerodynamic moment acting on the deck centre of rotation ( - C G ) .

The effect of windward shifts of the centre of rotation on the critical wind speed was investigated by Theodorsen and Garrick [3], from whom Fig. 3 is extracted, Fig. 3

Fig. 1. Photograph of ballast in l'orn~ of waler-tilled foil containers deployed oil the deck of the Humber Bridge during early stage,', of deck ercclion. British Bridge Builders [2].

A. Larsen/J. Wind Eng. Ind. Aerodyn. 72 (1997) 265-274 267

Q.8 -.5

I 1 I I I I I l i a !

--.4 ".3 --.2 -./ O .1 . .,Y a , ÷ , z a

.3 .4

, . . _ . . , ( ~ ) . _ 0 ,b, ._,~o, C .~ ) .~ .

Fig. 2. Physical principle underlying eccentric ballasting of deck.

Wind ~ >

Aerodynamic Lift Force

================================================ :~ii:j:::i::iiii::ii::iil!i::i::ili::iii::~ B i

_ AC C G I C?- t

Eccentric ~""'""""''"'""""*""' " ............ M a , m '

Fig. 3. Section flutter wind speed as function of chordwise posit ion of centre of gravity and elastic axis, adopted from Theodorsen and Garrick [3].

gives the non-dimensional flutter speed v/bco, as function of x~ the non-dimensional horizontal position of C G relative to the elastic axis for a "flat plate" airfoil of semi-chord b. Mass parameter and frequency ratio of the section are not unlike those of a suspension bridge during early stages of erection. Of particular interest is the curve a = 0 (elastic axis at mid-chord) found on graph (b). It is observed that the critical wind speeds increase almost exponentially for increasing windward (negative) shifts of CG. For x~ = - 0.05 the critical wind speed of the airfoil section is increased by approximately 33% where as x~ = - 0.1 ensures a 300% increase! x~ = - 0.05 may be obtained by adding ballast corresponding to 5.3% of the section mass at the windward section edge, whereas x~ = - 0.1 is secured by adding 11.1% of the section mass at this point.

F rom the discussion above, it is obvious that eccentric ballasting provides a means for enhancement of the flutter wind speed of bridge decks during erection. Extension of standard bridge flutter routines to accommodate this aspect is outlined below.

26S t . La~:s'ot,I. Wiml En£,. Ind. h, rodt'n. 72 H997) 265 274

3. Mathematical model

The mathematical model of lhe bridge deck is identical to the model adopted by Simiu and Scanhtn [4], Section 6.5. The deck is assumed to move as a rigid body with two degrees of freedom: the vertical displacement (bending) being denoted by h, and the torsional displacement (twist) by ~. Reference is made to Fig. 4. The deck section is assumed to be symmetrical with respect to the centreline, hence the elastic axis EA is located at mid-chord. The centre of gravity ( 'G may be displaced horizontally from EA at a distance e. The deck section of unit length is characterised by a mass m (kg/m) and a mass moment of inertia I (kg m2.m) about CG. Vertical and torsional eigenfrequencies are denoted by (,),, and ~,~ and the corresponding damping ratios are denoted by (~, and ~ relative-lo-critical.

With the definitions given above, the equations of motion of the deck section CG become

~ " ~ ,')i,h] I l ' : ( 2 B t m[ / i + e~ + -_~j,~,#,h ::: 2t)

{la)

+ (Ib)

where K = soB/U is the reduced fiequency relative to deck width and wind speed. The right-hand side of Eqs. (llt) and (I b)is the commonly accepted linear repres-

entation of measured motion-induced aerodynamic forces, denoted flutter coefficients o1" aerodynamic dcriwltives. The particular lbrm inw)lving a total of eight coefficients was proposed by Scanlan [5] as a logical extension of the "six coefficient" formulation originally proposed by Scanlan and Tomko [6]. Note that the dynamic head multi- plied to the sum of the flutter coefficients (right-hand side of Eqs. [ 1 a) and ( I bt is larger

U

Wind

- B B 2 2

e

EA CG

h

~ :kh = (~h2xm

k(z= co0Z x I 1~

Fig. 4. Degrees of frcedom, structural properties and sign con~cnlio~] adopted h)r Ihc mathematical model.


by a factor of 2 as compared to Scanlan [5], but in accordance with the original definition proposed by Scanlan and Tomko [6].

The solution of differential Eqs. (1 a) and (1 b) is based on the observation, that h and are harmonic in time with a common frequency co at the critical wind speed Uc for

onset of flutter. Introduction of time complex harmonic motions h = h exp(icot), :~ = ~ exp(icot) (i = imaginary unit) and a the flutter frequency ratio X = co/co, yields the following set of complex algebraic equations:

- - 2i(hX + PB2xzH* - ,j~B21 2 Ty~ h X: 1 -- m I ~ - - - A / - / l i b

+[_BX ~ "PB:--~-* PB:x:u,7: --I~--A /-12- m 3J , (2a)

7 7 ~ ~ j~ B 4 +[_X2_2i(:~X. ÷ k%/(co~=~2 - '~-~'PBCvz'*-~-XZA*]+,2 . (2b)

Setting the determinant of Eqs. (2a) and (2b) equal to zero results in a complex polynomial in X of degree four, which yields the real and imaginary part of the flutter determinant:

Real part:

P {a ' u* * *-- * * A~H* P B4A , 2 B 6 / m I 3 - ~ - - ~ 3 " ' 4 - A2H1 A 4 H 3 4-

I ~ 2 m e B , me 2 epff ] -- X4 - - 7 H3 + ~ [ - + ~ A~ (*)

B 4 + X3[2~CO~pB2H* + 2(h~-A *]

L " o) h m

-- - - - - 4 g h [ : - - 1 - - -- - -

\ cob / O)h I m \c%/ d

+ X. 0 + . (3a)

Imaginary part:

pZB4 , . , ] x3FpB2H*L~- 1+ PB4A*I 2 + 'm--~([ A2H4 + A~H* -- A'H* -- A~H*)

3[pB2meBH * PBCBA* ] (*) - X ~- 7 - 2 + i

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.\,2 ~,, ~')~ ~,, P B2 -~, P B4 , ] . ~ ; - , , , X ~ - - - H * - _¢,, A 3

( ' ) h - I~l T J

i;i \~;,;~ ' i ,47~ 4 =;,d, ¢'),,I/ + u),,J 0. (3b)

Comparison of the llutter equations given above to Eqs. (6). (5) and (9) presented by Simiu and Scanlan [4] reveals thai the shift in centre of rotation e appears in the l\)urth-order term {real pa r t ) and in the third-order term (imaginary part), here indicated by (*). In addilion the "new" derivatives A~. and H* extends the fourth and the second-order terms of the real equation and the third- and the second-order terms of the imaginary equation.

The solution of the extended llutter determinant proceeds as outlined by Simiu and Scanhm [4] by plotting curves corresponding to the roots of the real and imaginary parts Eqs. (3a)and {3b) of the ltutter determinant as function of the non-dimensional wind speed 1_; (,)B. The inlcrsection point between the real and imaginary root curves defines the critical wind speed as

4 . E f f e c t o f b a l l a s t i n g o n s t r u c t u r a l p a r a m e t e r s

The extended flutter determinant predicts the critical wind speed for onset of flutter once the relewml structural characteristics and the shift of CG relative to EA (centreline) arc known. These quantities are easily calculated once the amount of additional ballast per unil span Am and its lateral position x have been decided upon.

Introducing the ratio of ballast to section mass:

Am I~ (51

II1~)

yields the lollowing expressions lor determination of input to the flutter routine:

. . . . ~ - - . (6)

m m~(l 4 /d. (7)

/ 1

")" ~)""~/14 /i" (9)


~,~ = COco 1 + m o x 2 / I o [ ( 1 - (g/(1 +/~)2)p + (#/1 + g)2], (10)

where too, Io, COho and e&o refer to the deck section without ballast and m, I, e)h and ~,J~ include the corrections necessary to cater for addit ion of eccentric ballast.

5. Numerical example

The extended flutter routine detailed above was applied to a particular erection phase of a 1624 m main-span suspension bridge box girder. Flutter predictions were carried out using measured aerodynamic derivatives obtained from section model tests reported by Reinhold et al. [7] and reproduced in Fig. 5 for reference. For comparison, flutter calculations were also carried out using "flat plate" aerodynamic derivatives derived from the Theodorsen circulation function as given by Scanlan [5]. Structural parameters relevant to the numerical example are given in Table 1 below.

H* A*

6

4 ~ H2* 2 2 U

,-,

H1 0

-10 I ~ H3* -121 -1

AZ=o~ A3

Fig. 5, Measured aerodynamic derivatives for trapezoidal box girder section: left, H* derivatives (vertical motion); right: A* derivatives (torsional motion).

Table 1 Structural parameters used in numerical example

Degree of freedom Inertia properties Eigenfrequencies Damping rel-to-crit

Vertical h mo= 17.84 103 kg/m fho = 0.099 Hz ~h = 0.0008 Torsional ~ lo = 2.17 106 kg m2/m ,1;,o = 0.186 Hz ff=o = 0.008

272 ,t. Larxen ,1 ll"ind £n,~. hul. lerodvn. 72 (1997) 265 274

5. I. ()Ttic'a/ wind ,V)eed~'

Critical wind speeds obtained from predictions based on the two sets of aerodynamic derivatives are presented in Fig. 6 (left) as function of mass ratio tz. It is assumed that the mass ballast is located at a distance of 0.39B just inboard of the windward cable plane. Predictions arc compared to actual section model observations of the critical wind speed for It = 0.0, it 0 .11 and /~ 0.22. It is noted that each set of predictions flneasured and "fla! phtte'" aerodynamic deriwttives) yields an increase of the critical wind speed with increasing mass loading and thus increasing windward shifts of the deck centre of rotation t - CG). ttowever, only critical wind speed predictions based on measured aerodynamic derivatives (full line) agree well with direct observations on a wind tunnel section model with eccentric ballast (circle). Predictions based on theoretical "flat plate" aerodynamic derivatives (dashed line) overestimate the critical wind speed and in the present case yield non-conservative estimates of the amount of eccentric ballast needed to meet a specified critical wind speed. Fig. 6 (left) demonstrates thal the gab between the "fiat plate" prediction and the bridge deck prediction grows with increasing ballast mass ratio. A qualitative explanation for this observation is l\mnd in the differences between the bridge deck and the 'Tlal plate" derivatives. Examination of the real fourth-order and the imaginary third-order terms in the flutter determinant Eqs. (3a) and (3b) indicates that the critical wind speed in ballast condition becomes sensitive to H*, fI* and in particular to A*. A.~ as e 'B exceeds meh l by an order of magnitude. Comparison of A* and .4* for the bridge deck and the "flat plate" reveals substantial differences in these derivatives.

Critical wind speed Uc ( m / s )

70

60

50

40

3O

20

AD's / flat plate /

/ / Uc

/" Measured / Q

t / ' f ~ - )

ADs / ~ Measured

! Wind , i

0.39 x B ~, Mass Ratio /~

0.1 0.2 03

Critical wind speed Uc ( m / s )

Wind

o H9.1 . . . . . . . . . . . . |

o Wind Screen z~ H9.2

'Wind Screen Added

Nose down i i i

- 3 - 2 1

L

50

40

3O

Nose up a ( deg ) i

0 1 2 3

Fig. 6. Predicled and measured crilical wind spccd~, as iuncHon of ballast/deck mass ratio (lefH. Critical wind speed of bridge m service condition as function of angle of incidence Iright}.


5.2. Angle of incidence

Eccentric ballasting of a suspension bridge deck leads to an angular deflection of the section relative to horizontal. Hence, the wind flow meets the deck under a certain angle of incidence. Assuming that the wind flow is horizontal, the angle of incidence for the prototype bridge or a section model equivalent may be estimated as follows:

mo#xg 1 2 , (21) ~2- io~O2o 2121

where 9 is the gravitational constant, 2~ is the geometric scale and 21 is the frequency scaling. In case static rotational deflections are to be kept invariant from model to

prototype, Froude scaling is required, i.e., 21 = 1/x/~. In the wind-tunnel section model tests quoted above 2~ = 80 and 21 --- 4.96. Hence,

the static rotation for ballasted section is calculated as ~ = - 1.37 + for # = 0.11 and = - 2.75 for # = 0.22 which agree very well with experimental observations. The

predictions of critical wind speeds given in Fig. 6 (left) are based on aerodynamic derivatives obtained for ~ = 0 ° deg. Hence, the aerodynamics may not be quite identical to the actual load conditions of the model tests. Model tests of the deck in the "in service" condition (torsional/vertical frequency ratio = 2.7) indicated that the critical wind speed was virtually unaffected by the section angle of incidence for 0 ° < ~ < - 3 °, as shown in Fig. 6 (right). It may thus be assumed that measured

aerodynamics for ~ -- 0 ° are representative for the deflected (ballasted) conditions in the present case. Ideally aerodynamic derivatives should be obtained for a representative range of deck rotations including the unballasted case.

6. Conclusion

The present paper extends a standard bridge deck flutter routine given in the wind engineering literature in two ways: (1) inclusion of the full set of eight aerodynamic derivatives and (2) inclusion of a mathematical model description of eccentric ballast. Numerical calculations based on measured aerodynamic derivatives have been compared to direct measurements of the critical wind speed of a section model under identical ballast conditions. Good agreement between predictions has been demon- strated despite minor differences in the section angle of incidence. The present development is based on the assumption that the structural dynamics are full)' represented by a two-dimensional deck section, i.e. the mode shapes of the deck and the main cables are identical. Extension to the three-dimensional case where the main cables are substantially longer and have mode shapes different from the deck may be carried out within the framework of conventional modal analysis theory, This extension would be required for analysis of the early stages of deck erection of suspension bridges.

References

[1] F. Brancaleoni, The construction phase and its aerodynamic issues, in: A. Larsen (Ed.), Aerodynamics of Large Bridges, A.A. Balkema, Rotterdam, 1992.

274 .I. Larscn .I Wind t:'nk,, lml .4erodt'n. 72 (1997) 265 274

[2] British Bridge Builders, Bridging Ihc Itumber, ('erialis Press, York, (1981). [3] Th. fheodorsen. I.E. Garrick, A flleoretical cnd experimental investigation of the fltlttcr problem.

NACA Reporl No. 685. Langt~ Field, Virginia, 193N, [4] 1{. Simiu, R.H. Scanhm, Wind cl][~2cls Oll ',;IrtlCltlrcs, 2nd cd.. Wiley ln/erscience, New York, 1986. [5] R.H. Scanlan, Problemalics m wind l\~rcc models for bridge decks. ASCE ,1. Eng. Mech. 119 {1993)

1353 1375. [6] R.I|. Scanlan, .1,.I. Tomko. Airfoil ~md bridge dcck l]ulter deriwflixcs, ASCE J. Mech. Div. EM6,

Proceedings AS('I!, New York 17] f.A. Reinhold. M. Brinch, ,,\a [)anlsgailld. Wind lttll]lCl le';ts l'of ihc Great Belt East Bridge, m: A.

Larscn (Ed.). Acrodynamics of large bridges. ,\.A. Balkema. Rotterdam. 1992.

prediction of aeroelastic stability of suspension bridges during erection

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