A FORM OF AERODYNAMIC ADMITTANCE FOR USE IN BRIDGE AEROELASTIC ANALYSIS

Journal of Fluids and Structures (1999) 13, 1017}1027 Article No.: j#s. 1999.0243, available online at http://www.idealibrary.com on

A FORM OF AERODYNAMIC ADMITTANCE FOR USE IN BRIDGE AEROELASTIC ANALYSIS R. H. SCANLAN AND N. P. JONES Department of Civil Engineering, Johns Hopkins University, Baltimore, MD 21218-2686, U.S.A. (Received 22 September 1998 and in revised form 26 April 1999)

This paper returns to, and addresses, the question of identifying the nature of aerodynamic admittance in relation to extended-span bridges in wind. Theoretical formulations for the sectional aerodynamic forces acting upon the deck girder of a long-span bridge have conventionally been composed of the sum of two kinds of terms: aeroelastic terms and bu!eting terms. The former employ frequencydependent coe$cients (&&#utter derivatives'') associated with sinusoidal displacements of the structure, while the latter have typically been expressed in quasi-static terms with "xed lift, drag and moment coe$cients. This inconsistency of formulation has required that at some point the bu!eting terms, functions of gust velocity, be adjusted to a more compatible form through the introduction of the so-called aerodynamic admittance factors that are frequency-dependent. The present paper identi"es a form of these several section-force factors as functions of the #utter derivatives themselves. ( 1999 Academic Press

1. INTRODUCTION AT THE PRESENT TIME EXTENSIVE theory exists for forecasting the aeroelastic response of long-span bridges. Data in support of this theory are available in three categories: (i) static and motional force coe$cients measured on a wind-tunnel section model of the deck girder; (ii) computed natural bridge modes and frequencies of the whole structure; and (iii) spectral and coherence descriptions of the cross-wind. As has been repeatedly pointed out in the literature (Scanlan & Tomko 1971; Scanlan, BeH liveau & Budlong 1974; Scanlan, Jones & Singh 1997; Simiu & Scanlan 1996), transferring the application of theoretical thin-airfoil motional force coe$cients to the blu! bodies presented by bridge deck sections is basically incorrect from both mathematical and physical viewpoints. Thin airfoil dynamic force theories depend on unique circulation functions [such as those of Theodorsen (1935) and Sears (1941)] that cannot be realized for blu! bodies that experience separated #ow. It is therefore taken here for granted that the common use of the Sears admittance function in the context of bridge deck bu!eting can, at best, be viewed only as a suggested approximation. The Sears admittance function represents the dimensionless theoretical frequency-dependent force spectral level of a thin airfoil, with wholly attached #ow, penetrating a vertically oscillating gust "eld. While the use of the Sears function by Liepmann (1952) in the airfoil context is technically correct, that by Davenport (1962) in the bridge context is not. The present paper "rst brie#y reviews the main elements in the description of the aerodynamic forces, emphasizing in particular the relation between a form of aerodynamic admittance and the #utter derivatives. How these relations enter the consequent aeroelastic analysis is then suggested. 0889}9746/99/101017#11 $30.00

( 1999 Academic Press

1018

R. H. SCANLAN AND N. P. JONES

2. CONVENTIONAL SECTIONAL AERODYNAMIC THEORY In the case of a bridge deck section with degrees of freedom h (vertical), p (lateral), and a (twist) it has become conventional to separate the associated time-dependent sectional force vectors into aeroelastic (ae) and bu!eting (b) contributions. Thus, vertical lift: ¸"¸ #¸ , (1) ae b horizontal drag: D"D #D , (2) ae b moment: M"M #M , (3) ae b where, for sinusoidal motion, the aeroelastic terms involve, in commonly used form (Sarkar et al. 1994), up to six frequency-dependent aeroelastic or #utter derivative terms H*, P*, i i A* (i"1, 2, 6) within each force component: i 1 hQ BaR h pR p ¸ " o;2B KH* #KH* #K2H*a#K2H* #KH* #K2H* , (4) ae 2 1; 2 ; 3 4B 5; 6B

C C C

D

D

1 pR BaR p hR h D " o;2B KP* #KP* #K2 P*a#K2 P* #KP* #K2P* , ae 2 1; 2; 3 4B 5; 6B hQ BaR h pR p 1 M " o;2 B2 KA* #KA* #K2 A* a#K2 A* #KA* #K2 A* 1; 2 ; 3 4B 5; 6B ae 2

(5)

D

(6)

in which o is the air density, ; the mean cross-wind velocity, B the deck width, K"Bu/;, and u is the circular frequency of oscillation. This large array provides in advance for all possible motional contributions. Experimental evaluation brings out which of the #utter derivatives are important and which are not. Hence some formulations have been limited to fewer terms. The bu!eting forces have been written conventionally (Scanlan & Jones 1990) in quasistatic terms: 1 u w vertical: ¸ " o;2 B 2C¸ #C@¸ , (7) 0 ; 0; b 2

C C C

D

D D

u w 1 horizontal: D " o;2B 2CD #C@D , 0; 0; b 2

(8)

u w 1 , moment: M " o;2B2 2CM #C@M 0; 0; b 2

(9)

where C , C , C represent force coe$cients, C@ , C@ , C@ their slopes versus wind vertical L D M L D M attack angle a, and u and w the along-wind and vertical gust velocity components, respectively. In this formulation C and C are in perpendicular and horizontal directions, L D respectively. This conventional writing of the bu!eting forces implies association of a unit aerodynamic admittance factor with each force component. This simpli"ed, useful, but often incomplete formulation will be modi"ed at a later point. 3. HEURISTIC REEXAMINATION OF SECTIONAL FORCES The basic conceptual formula for sectional lift at any instant is 1 ¸" o;2 (t) BC (t) , r L 2

(10)

1019

AERODYNAMIC ADMITTANCE IN BRIDGE ANALYSIS

where ; and C may be considered variable, ; representing the relative horizontal wind r L r velocity, lift ¸ being understood as vertical. If the structural section has displacement components h vertical and p horizontal, and the gust velocity components are u and w, then one may write ; ";#u!pR , (11) r while C may be represented as a &&base'' value C¸ plus an increment due to a small angular 0 L change w!h/;: C (t)"C¸ #C@¸ 0 0 L

A B w!hQ ;

(12)

for C@¸ "dC¸ /da, where C¸ accounts for all lift contributions*such as mean, steady-state 0 0 0 or signature (internal structure-induced) e!ects*not due to the explicit horizontal and vertical causes mentioned. Thus,

C

A BD

w!hQ 1 ¸" oB [;#u!pR ]2 C¸ #C@¸ 0 0 ; 2

,

(13)

which, if products of small quantities relative to ; are neglected, becomes

C

A B

A BD

w!hQ u!pR 1 #2C¸ ¸" o;2B C¸ #C@¸ 0 0 0 ; ; 2

.

(14)

This then represents a base value plus the variable lift contributions ascribable to the speci"c horizontal and vertical e!ects mentioned. It is important to emphasize the concept of gusting that is inherent in the above formulation. Both the horizontal and vertical velocities u, w are assumed in this writing to extend unchanged spatially over the full height or width, respectively, of the bridge deck section in question. In other words, the assumption implicit in these expressions is that, over the extent of the deck section geometry in either direction, vertical or horizontal, the gust velocity is completely spatially coherent. This is a conservative assumption regarding developed force values. It could, if desired, be modi"ed through the introduction of coherence postulates, which will not be done here. In equations (4)}(6) the assumed motion of h, p and a is purely sinusoidal. If this is represented in the form e*ut then the particular freedoms h and p may be represented in terms of hQ and pR as hQ /iu, pR /iu, respectively. The time-varying terms of equation (14) may now be compared to the u, w, hQ and pR terms in equations (1), (4) and (7), according to which the terms associated with vertical and horizontal contributions to lift are, together:

C

D

1 hQ pR u w ¸ " o;2B K[H*!iH*] #K[H*!iH*] #2C¸ #C@¸ . 0 0 hp 2 4 ; 6 ; 1 5 ; ;

(15)

This suggests that in equation (14) the following equivalences are appropriate: 2C¸ "!K[H*!iH*] , (16) 0 5 6 (17) C@¸ "!K[H*!iH*] , 0 1 4 These recognize the inherent phasing and frequency dependence required of C¸ , C@¸ . 0 0 By analogous reasoning applied to drag and moment expressions, the following equivalences are also suggested: 2CD "!K[P*!iP*] , 0 1 4

(18)

1020

R. H. SCANLAN AND N. P. JONES

C@D "!K[P*!iP*] , 0 5 6 2CM "!K[A*!iA*] , 0 5 6 C@M "!K[A*!iA*] , 0 1 4 This conversion alters the bu!eting force terms to the form u w ¸ b "!K [H*!iH*] !K [H*!iH* ] , 5 6 ; 1 4 ; 1 o;2B 2 D u w b "!K [P*!iP*] !K [P*!iP*] , 1 4 ; 5 6 ; 1 o;2B 2 M u w b "!K[A*!iA*] !K[A*!iA*] 5 6 ; 1 4 ; 1 o;2B2 2 From these, the autospectrum S of lift becomes LL S LL "K2 [H*2#H*2]S #K2[H*!iH*][H*#iH*] S 5 6 uu 5 6 1 4 uw [1 o;B]2 2 #K2[H*#iH*][H*!iH*]S #K2[H2#H2] S , 5 6 1 4 wu 1 4 ww where S is the cross-spectrum of u and w. uw Those for drag and moment are constituted similarly:

(19) (20) (21)

(22) (23) (24)

(25)

S DD "K2[P*2#P*2] S #K2[P*!iP*] [P*#iP*]S 1 4 uu 5 6 1 4 uw [1 o;B]2 2 #K2[P*#iP*][P*!iP*] S #K2[P*2#P*2] S , (26) 5 6 1 4 wu 5 6 ww S MM "K2[A*2#A*2]S #K2 [A*!iA*] [A*#iA*]S 5 6 uu 5 6 1 4 uw [1 o;B2]2 2 #K2[A*#iA*] [A*!iA*]S #K2[A*2#A*2] S , (27) 5 6 1 4 wu 1 4 ww where S is the cross spectrum of u and w. uw If earlier formulations (22)}(24) are referred to, and the cross-spectra of u and w are neglected, the results corresponding to expressions (25}27) above were S LL "4C2¸ s2 S #(C@ )2s2 S , (28) 0 L uu L L{ ww [1 o;B]2 2 S DD "4C2D s2 S #(C@D )2s2 S , (29) drag: 0 D uu 0 D{ ww [1 o;B]2 2 S MM "4C2M s2 S #(C@M )2s2 S , (30) moment: 0 M uu 0 M{ ww [1 o;B2]2 2 where frequency-dependent correction factors s2, aerodynamic admittances, have been incorporated to account for frequency dependence. This formulation is seen to be superseded by equations (25)}(27). In fact the notion of aerodynamic admittance is clari"ed by this process; it suggests that a concept of aerodynamic admittance can be viewed as de,ned in terms of -utter derivatives, which imply the following de"nitions: lift:

4C2s2"K2[H*2#H*2] , 5 6 L L * * 2 4C2 s2 "K2[P #P 2] , 1 4 D D

(31) (32)

AERODYNAMIC ADMITTANCE IN BRIDGE ANALYSIS

1021

Figure 1. Representative aerodynamic admittances: (1) unitary (for quasi-static gust forces); (2) example corresponding to H* of Figure 2; (3) classic Sears function. 1

4C2 s2 "K2[A*2#A*] , 5 6 M M (C@ )2s2 "K2[H*2#H*2] , 1 4 L L{ (C@ )2s2 "K2[P*2#P*2] , 6 5 D D{ * (C@ )2s2 "K2[A 2#A*2] , 4 1 M M{

(33) (34) (35) (36)

These formulas suggest relationships that have not to-date been widely demonstrated by available data. Two brief illustrations may, however, be given. If H* is proportional to 2n/K 1 (as is sometimes the case) and H* is negligible (also reasonable), then the admittance s2 has 4 L{ a constant unit value as in quasi-steady theory; (see curve 1, Figure 1). If, as a further example, H* is as illustrated in Figure 2 for a realistic case for which C@ "4)11 per radian, L 1 with H* again negligible then s2 evolves as shown in curve 2, Figure 1, which evidences 4 L{ a trend resembling somewhat that of the Sears function, curve 3 of Figure 1. The point of the present discussion is to open up a promising vector of investigation. 4. TIME-DEPENDENT ANALYSIS OF SECTIONAL AERODYNAMIC EFFORTS The discussion in this section essentially reviews and arrives at the same results as the analysis of the preceding section but from a transient-aerodynamic viewpoint. ¸ift as a representative case will be examined "rst. Consider a deck section undergoing a vertical velocity hQ (t) and simultaneously the impact of a vertical gust w(t) . The net e!ective vertical wind angle of attack h is then w!hQ h" ;

(37)

1022

R. H. SCANLAN AND N. P. JONES

Figure 2. Typical #utter derivative H* for bridge deck. 1

and the corresponding transient lift is ¸(s)"1 o;2BC@ I(s), 2 L

(38)

where s";t/B is dimensionless time, C@ "dC /dh and I(s) is the lift-growth integral: L L

P

I(s)"

s ~=

'(s!p) h@(p) dp

(39)

in which h@(p)"dh/dp and '(s) is an appropriate &&indicial'' function. 4.1. THE FLUTTER CASE, w"0 Information on ' may be obtained from the standard #utter case in which w"0, conventionally written (4) as hQ 1 ¸" o;2B [KH*!iKH*] , 1 4 ; 2

(40)

where H* is a #utter derivative and hQ is purely sinusoidal. 1 With the change of variable s!p"q, I(s) may be written as

P

I(s)"

=

0

P

'(q) h@(s!q) dq"

=

0

h(s!q) '@(q) dq

(41)

in which '@(q)"d'/dq so that h"h e*Ks: 0 ¸(s)"1 o;2BC@ ['(0)#'@] h eiks"1 o;2B [KH*!iKH* ] h e*Ks , L 0 1 4 0 2 2

(42)

1023

AERODYNAMIC ADMITTANCE IN BRIDGE ANALYSIS

where

P

'@"

=

0 is the Fourier transform of '@(q)"d'/dq. This yields the net relation

'@(q) e~*Kq dq

(43)

C@ ['(0)#'@]"K[H*!iH*]. L 1 4

(44)

4.2. THE BUFFETING CASE, hQ "0 We now consider the alternate case in which h"w/; and hQ "0. In this case

C

P

D

=w(s!q) w(s) 1 # '@(q) dq , ¸(s)" o;2BC@ '(0) L ; ; 2 0 of which the Fourier transform is wN 1 wN 1 ¸M " o;2BC@ ['(0)#'@] " o;2B [KH*!iKH*] , L 1 4 ; 2 ; 2

(45)

(46)

Multiplying ¸M by its complex conjugate, it can immediately be demonstrated that the autoPSD of lift is S "(1 o;B)2 K2 [H2#H2] S . 4 ww 1 LL 2 Recalling the quasi-steady expression for bu!eting lift, ¸ w "C@ L; 1 o;2B 2

(47)

(48)

which leads to the lift spectrum S LL "(C@ )2 S s2 (49) L ww L{ (1 o;B)2 2 as &&corrected'' by the admittance factor s2 , it can be seen that the latter is de"ned [see L{ equation (25)] by (50) s2 "K2 [H2#H2]/(C@ )2 , L 4 1 L{ where the relation of s2 to '@ is L{ s2 "['(0)#'@] ['(0)#'@*] . (51) L{ Clearly, the analogous expressions for admittances s2 in equations (31)}(36) are derivable by similar methods. The analysis of this section, while con"rming that of the preceding, further links aerodynamic admittance directly to its associated indicial function source. Connections of this character were underscored by Scanlan (1984, 1993). 5. SINGLE-MODE FLUTTER AND BUFFETING Although multi-mode analyses are now routinely performed (Jain et al. 1996; Katsuchi et al. 1998a, b), attention will be con"ned here to some remarks on the general process, with focus only on single-mode response.

1024

R. H. SCANLAN AND N. P. JONES

5.1. EQUATION OF MOTION This, with arbitrarily limited choice of #utter derivatives, is I [mG #2f u mQ #u2m ]"Q , i i i i i i i i

(52)

where I is the whole-bridge generalized inertia of mode i, and, i

G

1 KB Q " o;2B2l [H*Gh h #P*Gp p #A*Ga a ] mQ . i 2 1 ii 1 ii 2 ii i ;

P

#K2A*Ga a m # 3 ii i

$%#,

dx [Lh #Dp #Ma ] i i i l

H

(53)

and

P

Gq q " i i

$%#,

dx q2(x) [q "h , p or a ], i i i i i l

(54)

and where the bu!eting terms L, D, M will be described later. Note that this form conservatively implies full coherence of #utter derivative action along the span. Alternative choices are easily and routinely incorporated, as desired. 5.2. FLUTTER With damping restricted to only the principal elements the single-degree #utter criterion reduces to

C

D

1@2 4f I oB4l H*Gh h #P*Gp p #A*Ga a 5 i i 1# . A*Ga a 1 ii 1 ii 2 i i oB4l 3 ii 2I i

(55)

5.3. BUFFETING Rewriting equations (52) and (53) yields the form

P

o;2B2l dx mG #2c u mQ #u2 m " [Lh #Dp #Ma ] , i0 i i i i0 i i i i l 2I i $%#,

(56)

where damping f and frequency u have been replaced by c , u , their respective aerodyi i i i0 namically in#uenced counterparts: oB4l u2 "u2! u2A* Ga a i0 i 3 ii 2I i

(57)

oB4l 2c u "2f u ! u [H*Gh h #P*Gp p #A*Ga a ] . 1 ii 1 ii 2 ii i i0 i i 2I i

(58)

and

The Fourier transform of equation (56) is

P

o;2B2l dx [u2 !u2#2ic u u] mM " [L M h #DM p #MM a ] . i i0 i i0 i i i 2I l i $%#,

(59)

AERODYNAMIC ADMITTANCE IN BRIDGE ANALYSIS

1025

Multiplying equation (59) by its complex conjugate yields

C

D PP

o;2B2l 2 2I i where ( ) *"complex conjugate of ( ) and [(u2 !u2)2#(2c u u)2] mM mM *" i0 i i0 i i

$%#,

dx dx %(x , x , u) A B A B l l

(60)

%(x , x , u)"[L M (x , u) h (x )#DM (x , u) p (x )#MM (x , u)a (x )] A B A i A A i A A i A ][L M * (x , u) h (x )#DM * (x , u) p (x )#MM *(x , u) a (x )] . (61) B i B B i B B i B It is at this point that the newly introduced concept of admittance enters the analysis. The lift (L), drag (D) and moment (M) factors above depend on the horizontal and vertical (u, w) components of gusting and are written in their new forms [see equations (22)}(24)]: u w L"!K[H*!iH*] !K[H*!iH*] , 6 ; 4 ; 5 1

(62)

u w D"!K[P*!iP*] !K[P*!iP*] , 4 ; 6 ; 1 5

(63)

w u M"!K [A*!iA*] !K [A*!iA*] . 4 ; 6 ; 1 5

(64)

The details of gust analysis proceed from this point. What has been stated above constitutes an abbreviated overview of how the newly de"ned aerodynamic admittances enter the bu!eting problem. The details will not be pursued in the present paper. 6. SUMMARY AND CONCLUSION The present paper "rst derives and emphasizes the natural analytical relationships between corrective aerodynamic admittances associated with gust forces and certain well-known #utter derivatives. The roles and implications of these relationships are then suggested by brief remarks on the theoretical steps of #utter and bu!eting analyses. Particularly, the proper role of aerodynamic admittances as well-de"ned frequency-dependent functions intrinsic to the analysis, rather than arbitrarily imposed external correction factors, is emphasized. The paper opens new avenues of investigation in which the e$cacy of the proposed rationale for aerodynamic admittance can be more closely examined. ACKNOWLEDGEMENTS The authors wish to thank the reviewers of the original manuscript of this paper for helpful suggestions. REFERENCES DAVENPORT, A. G. 1962 Bu!eting of a suspension bridge by storm winds. ASCE Journal of Structures Division 88, 233}264. JAIN, A., JONES, N. P. & SCANLAN, R. H. 1996 Coupled #utter and bu!eting analysis of long-span bridges. ASCE Journal of Structural Engineering 122, 716}725. KATSUCHI, H., JONES, N. P. & SCANLAN, R. H. 1998a Multi-mode coupled #utter and bu!eting analysis of the Akashi}Kaikyo bridge. ASCE Journal of Structural Engineering 125, 60}70. KATSUCHI, H., JONES, N. P., SCANLAN, R. H. & AKIYAMA, H. 1998b Multi-mode #utter and bu!eting analysis of the Akashi}Kaikyo bridge. Journal of =ind Engineering and Industrial Aerodynamics 77 & 78, 431}441.

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R. H. SCANLAN AND N. P. JONES

LIEPMANN, H. W. 1952 On the application of statistical concepts to the bu!eting problem. Journal of Aeronautical Science 19, 793}800, 822. SARKAR, P. P., JONES, N. P. & SCANLAN, R. H. 1994 Identi"cation of aeroelastic parameters of #exible bridges. ASCE Journal of Engineering Mechanics 120, 1718}1742. SCANLAN, R. H. 1984 Role of indicial functions in bu!eting analysis of bridges. ASCE Journal of Structural Engineering 110, 1433}1446. SCANLAN, R. H. 1993 Problematics in the formulation of wind force models for bridge decks. ASCE Journal of Engineering Mechanics 119, 1353}1375. SCANLAN, R. H. & TOMKO, J. J. 1971 Airfoil and bridge deck #utter derivatives. ASCE Journal of Engineering Mechanics 97, 1717}1737. SCANLAN, R. H., BED LIVEAU, J. G. & BUDLONG, K. 1974 Indicial aerodynamic functions for bridge decks. ASCE Journal of Engineering Mechanics 100, 657}672. SCANLAN, R. H. & JONES, N. P. 1990 Aeroelastic analysis of cable-stayed bridges. ASCE Journal of Structural Engineering 116, 279}297. SCANLAN, R. H., JONES, N. P. & SINGH, L. 1997 Inter-relations among #utter derivatives. Journal of =ind Engineering and Industrial Aerodynamics 69}71, 829}837. SEARS, W. R. 1941 Some aspects of non-stationary airfoil theory and its practical application. Journal of Aeronautical Science 8, 104}108. SIMIU, E. & SCANLAN, R. H. 1996 =ind E+ects on Structures, 3rd edition. New York: WileyInterscience. THEODORSEN, T. 1935 General theory of aerodynamic instability and the mechanism of #utter. NACA Report 496, U.S. Advisory Committee for Aeronautics, Langley, VA, U.S.A.

APPENDIX: NOMENCLATURE A* i B D D Gq q i i H* i h h (x) i I i K ¸ L l M M n P* i p p (x) i Q i s S uu S ww S uw t ; u w x a a (x) i c i f i

twist #utter derivative deck width sectional drag force dimensionless drag factor, bu!et modal integral (q "h , p or a ) i i i i vertical motion #utter derivative sectional vertical displacement deck horizontal component, mode i generalized inertia of entire bridge Bu/; sectional lift force dimensionless lift factor, bu!et bridge span sectional moment dimensionless moment factor, bu!et frequency in Hz sway #utter derivative sectional lateral displacement deck sway component, mode i generalized modal force, mode i dimensionless time or distance, ;t/B power spectral density of horizontal (u) gust component power spectral density of vertical (w) gust component cross-power spectral density of u, w gust components time cross wind velocity horizontal gust velocity vertical gust velocity spanwise coordinate along deck sectional rotation deck twist component, mode i aerodynamics-in#uenced total damping ratio, mode i mechanical damping ratio of mode i

AERODYNAMIC ADMITTANCE IN BRIDGE ANALYSIS

h m i % o q ' '@ s2 u u i0 u i

general wind angle of attack generalized coordinate of mode i gust cross-coherence integrand air density integration variable indicial force-growth function d'/ds aerodynamic admittance factor circular frequency of oscillation (#utter) circular frequency of oscillation (bu!et) natural circular frequency of mode i

1027