Consistent Numerical Model for Wind Buffeting Analysis of Long-Span Bridges

Transcription

1 Consistent Nmerical Model for Wind Bffeting Analysis of Long-pan Bridges Dorian JANJIC Technical Director TDV GesmbH Graz, Astria Heinz PIRCHER Director TDV GesmbH Graz, Astria mmary The bffeting analysis of bridge strctres considers both, the aero-elastic behavior of the strctre and the ind loading correlation This calclation is commonly performed in the modal space and in the freqency domain It incldes aerodynamic damping and stiffness effects de to the strctral movement cased by the ind flo The aero-elastic parameters of the cross-sections (drag, lift, pitching moment and derivatives, sally determined in ind tnnel tests) are the basis of this type of analysis The non-linear dependency of these parameters on the ind direction mst be acconted for dring both, the static mean-ind and the bffeting analysis The ind profile is characterised by the mean ind velocity and the flctation (trblence) velocity here the height profile of the ind velocity is assmed to follo an exponential expression The stochastic natre of ind loading (longitdinal, vertical and lateral components of the flctation velocity) is acconted for by sing poer spectra in the freqency domain The spatial distribtion (coherence) of the ind loading is commonly determined from an exponential decay fnction of the freqency and the geometry of the strctre This paper reports on performing a bffeting analysis in a commercial bridge design compter program The implementation of the bffeting analysis in a bridge design program means that all non-linear effects that may have taken place prior to the ind event can be inclded in the calclation This allos a complete global analysis of bridge strctres and has been applied to the bffeting analysis of the Rach Mie Bridge in Vietnam and the henzhen Western Corridor as ell as the tonectters Bridge in Hong Kong that serve as application examples in this paper Keyords: bffeting analysis, aero-elastics, dynamics, compter analysis, bridge design 1 Introdction As dynamic effects de to ind loading, especially bffeting, often inflence the design of longspan bridges considerably, the capability of performing a bffeting analysis as implemented in a commercially available bridge-design softare package called RM004 (TDV 004) This featre is no flly integrated ith all the other fnctions of the softare package Mch effort has recently been devoted to frther improve the dynamic fnctionality in RM004 In addition to the previosly available modles that inclded all standard procedres for earthqake design (eigenmode analysis, response spectrm analysis and time history based on modal analysis) a variety of ne featres have been added The ability to perform bffeting analyses is among them pectral definition of the ind loading reqires that the analysis mst inclde all freqency ranges (from zero to freqency infinity) Both, the strctral and the aerodynamic damping mst be considered in the analysis The strctral bffeting calclation can be performed in the freqency domain (modal space) or in the time domain (direct integration in time) The stochastic natre of the ind and the spectral definition of the ind loading clearly indicate that a soltion in the freqency domain is more appropriate The analysis covers all freqency ranges ith a consistent definition of the strctral damping and aerodynamic damping and stiffness trctral and/or aerodynamic nonlinearities are generally reqired to be inclded in the calclation of longer-span bridges and therefore time domain approaches become more attractive again trctral nonlinearities can be covered in the time domain, bt several difficlties arise: the time history of the ind loading is nknon, the definition of the strctral damping is restricted, and

2 the nmerical ct-off of the high-freqency response depends on the chosen time steps This paper reports on the implementation of a soltion to the bffeting isse in the bridge design softare package RM004 [1] This implementation is carried ot in the freqency domain Bffeting Analysis General Description The strctral bffeting calclation is performed in the modal space and in the freqency domain It incldes aerodynamic damping and stiffness effects de to strctral movement cased by the ind flo All comptations are based on the tangential stiffness of the strctre at a given moment in time the strctre nder permanent loading and mean ind - ensring the inclsion of all nonlinear effects hich may have taken place prior to the stochastic ind event The basis of the implemented bffeting analysis fnctions is the aero-elastic cross-section parameters of the strctral members (drag, lift, pitching moment and derivations) as determined in ind tnnel tests The dependency of these parameters on the ind direction is atomatically acconted for dring the static mean-ind analysis and the bffeting analysis in RM004 The ind profile is characterised by the mean ind velocity and the flctation (trblence) velocity here the height profile of the ind velocity is assmed to follo an exponential expression The stochastic natre of ind loading (longitdinal, vertical and lateral components) is acconted for by sing poer spectra in the freqency domain (Kaimal, Karman, Harris, Davenport or individally defined) Narro band analysis as ell as broad band analysis ith fll modal correlation has been implemented The spatial distribtion (coherence) of the ind loading is determined from an exponential decay fnction of the freqency and the geometry of the strctre Peak reslts for design are estimated combining mean ind analysis reslts ith root-mean-sqare bffeting analysis reslts sing the corresponding peak factors These reslts are provided in sch a ay that they can be sed for post-processing like any other analysis reslts prodced by RM004 Fndamental aerodynamic characteristics of the bridge deck The aerodynamic coefficients that are needed for all sed cross sections are sally obtained from sectional ind tnnel tests (sally deck section) or from pblic literatre These aerodynamic coefficients lead to the derivation of crves for drag, lift force and pitching momentm Aerodynamic coefficients C D, C L and C M depend on the ind attack angle a Wind attack angle is an angle beteen a projection of the ind vector in the element cross plane and a chosen cross plane axis This axis is axis Z in Figre 1 belo a + rz r r z y Z vrel Y r y Lift Moment Drag Eqation 1 Figre 1: Velocity of the strctre and ind flctation Drag, lift and moment coefficient are sally given in non-dimensional form They are normalised ith some reference length and area (sally idth B and area B²) from the sectional model teady state coefficients C D, C L and C M are sed to calclate forces on strctral elements 4 Bffeting forces acting on the bridge deck ( α ) D α 1 ρ B C D ( α ) L α 1 ρ B C L C M ( α ) M ( α ) 1 ρ B It is necessary to take into accont the possible movement of the strctral sections relative to the ind flo in the final expression for the bffeting forces Only the simple case of the mean ind profile direction vector lying completely parallel to the element local +Z axis ill be discssed here Velocity of the strctre in the local Z direction ( r z ) and in the local Y direction ( r y ) together ith a tisting arond local X axis ( ϑ x ) ill change relative to the ind velocity vector and relative to the ind attack angle

3 Figre 1 shos the additional ind attack angle α de to the velocity of the strctre and ind flctation components Taking into accont tisting rotation arond local X axis ( ϑ x ) total change of the ind attack angle α can be approximated as shon in Eqation " " ( α) + α, C ( α) C + α C, ( α) C C C D D D Eqation L L L r α ϑ + y Eqation x C C C " M M+ α M Linearization of the steady state drag, lift and moment coefficient is done arond the ind attack angle nder the mean ind loading (Eqation ) Wind bffeting forces are finally separated into static (mean ind), dynamic (ind flctation) and aerodynamic terms (aerodynamic damping and stiffness) tatic terms are covered in the non-linear static calclation Dynamic and aerodynamic terms ill be analysed in the freqency domain The final expression for the bffeting forces is shon in Eqation 4 1 ' 1 ' 1 ' BCD+ B( CD CL) BCDrz B( CD CL) ry BC D BC + Dϑx ' L ρ VC + ρ BC + B C + C ρ BC r + B C + C r + ρ BCLϑ x M 1 ' BCMϑx D ' ' L L L D L z L D y BCM tat 1 ' 1 ' BC M + BCM BCMrz + BCMry Dyn Damp Eqation 4 Mean ind profile - non-linear static calclation The first step of the analysis is the non-linear static calclation nder a given permanent loading and a chosen mean ind profile in a given direction All strctral static nonlinearities can be inclded by RM004 in this step The cable sagging effects, p-delta and large displacement effects, non-linear spring and/or bffer elements are combined ith an additional mean ind loading Even creep and shrinkage effects (additional strains, stress and displacement terms) are inclded in the analysis from time zero p to the ind event, provided the constrction history is knon tiff Wind velocity Peak RM time Figre : Wind velocity in time domain at one D point Height(m) Constant mean ind Wind flctation Wind velocity (m/s) Mean ind Figre : Wind velocity profile over the height Wind loading has a stochastic natre becase ind speeds vary randomly ith time This variation is de to the trblence of the ind flo Wind speed at one single D point can be separated into the mean ind speed and the flctation (trblence) term as shon in Figre Mean ind speed varies ith the height Different assmptions for this variation are available, and a sal assmption is some kind of exponential expression: (h) ref ( h/ h ref ) α Coefficient α is an exponent dependant pon roghness of the terrain and varies beteen 01 and 04 Dependency of the steady-state drag, lift and pitching moment coefficients on the ind attack angle are consistently inclded in this analysis step???? C C H C V C M ??( a [ ] Figre 4: Aerodynamic coefficients of bridge deck for L o ) C D C L RM004 takes interaction of the strctre and the mean ind loading into accont by recalclating the crrent position of each strctral element de to the mean ind, folloing each diagram (C D, C L, C M ) as shon in Figre 4 (Rach Mie Bridge) At each iteration step the mean ind loading is pdated and the overall eqilibrim is checked ith the general Neton-Raphson procedre

4 6 Eigenmode Analysis The next step is a determination of the strctral eigenmodes and eigenvales nder crrent total loading in the displaced strctral shape The state of one fictive strctral degree of freedom is shon in Figre 7 This state is prior to the trblence (flctation) ind loading in the nmerical analysis Instead of the linear strctral stiffness, the strctral tangent stiffness nder permanent and mean ind loading is sed by RM004, and the classical eigenmode analysis can be performed Load The soltion of [K T ] ω² [M] 0 gives eigenmodes and eigenvales This step is repeated for each ne KT Mean Wind Load crrent KT mean ind direction tage loads elf Weight 1 1 KT1 Tangent tiffness Displacement Total Displacements incremental Displacements The sage of the strctral tangent stiffness reslts in the best possible linearization for the sbseqent dynamic analysis steps in the freqency domain Figre : state of one strctral degree of freedom 7 Bffeting analysis in the freqency domain RM response Bffeting loading (flctation part) is defined as the nsteady loading of a strctre by velocity flctation in the oncoming flo The magnitde of velocity flctations is characterised by the non-dimensional qantity termed the trblence intensity The trblence intensity component I is defined in Eqation belo, here s i is the standard deviation of the flctation velocity component in each direction Eqation Trblence intensity generally varies over the height as shon in Figre 8 at the tonectters Bridge, Hong Kong Figre 6 Velocity flctation of the incoming flo is sally defined by ind poer spectra Empirical formlae for some ell-knon spectra implemented in RM004 are given in Eqation 6 f σ z 00 * n z ( f ) χ (1 1 χ ) + ( ) ( 1+ x ) 10 x 4 4 k z 1 * v nz Van der Hoven pectra f v ( f ) χv σ v (1+ 1 χ ) v Kaimal pectra 1 k 10 y v ( ) 1+ 9y Davenport pectra z 6 * n z ( f ) χ f σ (1+ 1 χ ) ( ) 6 k y 10 ( 1+ 10y) Eqation 6 The cross-spectrm beteen to points in space is generally complex-valed despite the fact that the single-point spectrm in either of the points is real-valed I s I v s v I s As a fll D model is being treated, this flctation is given separately by different ind poer spectra for longitdinal (), horizontal (v), and vertical () flctation component Different flctation poer spectra have been tested and implemented Most of the spectra are given in empirical form depending on integral scale lengths, mean ind velocity and freqency C ( ϖ ) ( ϖ) ( ϖ) χ ( ϖ), ij χ kk ( ϖ ) i Ckk( ϖ ) e, kk ( ϖ ) cx x + cy y + cz z ϖ π ( + i j ) Eqation 7 j The non-zero imaginary part accommodates the fact that, a distrbance occrring in a flid at one point, ill occr at the other point some later time

5 The absolte vale of the to-point cross-spectrm is generally expressed by means of the root cross-coherence fnction as shon in Eqation 7, here k, l, v or bscripts i and j refer to the points i and j in space, respectively The empirical exponential decay la is idely applied in ind engineering applications, here c l and l are the decay coefficient and the separation in the direction of l x, y or z coordinate Mltimode bffeting calclation is done in the freqency domain taking aerodynamics damping and stiffness into accont Narro band analysis as ell as broadband analysis ith fll modal correlation can be done In bridge engineering, here strctral damping is rather small, aerodynamics damping plays an important role as its order of magnitde is the same as that of the strctral damping The reslt from this calclation step is the RM envelope reslt for forces, stresses, displacement, velocity and acceleration 8 Prediction of the peak bffeting response The last step in the analysis is a prediction of the expected vale of the largest peak The peak response is taken as the sm of the mean ind response and an RM bffeting response increased by peak factors K(n) n² z dn 0 ν z n dn 0 1/ K n ( ln 1/ ν T ) (lnνt ), 1/ Eqation 8 Mode nmber Freqency (Hz) Aerodynamic damping Peak factor K(n) 1 0,1484 0,00899,18 0,1889 0,0146,68 0,1918 0,0111,61 4 0,197 0,010,409 0,4 0,0461,10 6 0,147 0,0409, ,07 0,01491, ,670 0,0066,496 Table 1: K(n) vales for the first 8 modes 9 Application examples sally it is assmed that the probability distribtion of the ind event may be of the Poisson type It can be shon that the peak factor K for sch a signal depends on its spectral density fnction and freqency n The approximation given in Eqation 8 is adeqate for se in practical calclations Vales for displaceme nt peak factor K(n) are sally in the range beteen and as is shon in Table 1 (tonectters Bridge, HK) Global analysis inclding ind-bffeting response has been sccessflly applied to the Rach Mie Bridge in Vietnam, the henzhen Western Corridor and the tonectters Bridge in Hong Kong This experience shos that ith a groing span ind events and non-linear effects become important design criteria in bridge engineering The Rach Mie Cable tayed Bridge (Project Engineers: Transport Engineering Design Inc, Hanoi) ith a span arrangement of 117 : 70 : 117 m comprises a prestressed concrete deck girder ith reinforced concrete pylons and sbstrctre Figre 7: trctral RM004 model, Rach Mie Bridge, Vietnam Figre 8:Eigenmodes mainly indced in bffeting response

6 A cable stayed bridge ith a main span of 10 m is on the Hong-Kong side of the henzhen Western Corridor (Project Engineers: Ove-Arp, Hong-Kong) Figre 10: Lateral bending moment Figre 9: trctral RM004 model, henzhen Western Corridor Figre 11: Vertical bending moment The steel girder is spported by an inclined pylon 18m high Wind bffeting responses are shon in Figre 10 to Figre 1 Figre 1: Normal force The tonectters bridge (Project Engineers: Ove-Arp, Hong-Kong) is a cable-stayed bridge ith a main span of 1018m, side spans of 98m, and a toers height of 90m The deck to the main span is a tin girder steel deck, hilst the side spans are of concrete, hich are to be bilt in advance of cable stay erection to conterbalance and stabilise the slender lighteight main span deck Wind bffeting responses are shon in Figre 14 to Figre 16 Figre 14: Toer bending moment Prefabricated cable stays are arranged in a laterally inclined fan arrangement to maximise the transverse and torsional stability of the main span and inclde the orld s longest bridge stay to date Figre 1: trctral RM004 model, tonectters Bridge, HK Figre 1: Deck bending moment Figre 16:Deck normal force 10 References [1] TDV GesmbH (004) RM04 Technical Description, Graz, Astria, 004 [] Emil imi and Robert Hcanlan -Wind effects on strctres, John Wiley & ons, Ne York, 1996 [] Erik Hjorth Hansen, Wind engineering lectre notes, The niversity of Trondheim, Noray, 1988