EFFECTS OF ERRORS IN FLUTTER DERIVATIVES ON THE WIND- INDUCED RESPONSE OF CABLE-SUPPORTED BRIDGES

Transcription

1 EFFECTS OF ERRORS IN FLUTTER DERIVATIVES ON THE WIND- INDUCED RESPONSE OF CABLE-SUPPORTED BRIDGES A Dissertation Presented by Dong-Woo Seo to The Department of Civil and Environmental Engineering in partial fulfillment of the requirements for the degree of Doctor of Philosophy in Civil Engineering in the field of Structural Engineering Northeastern University Boston, Massachusetts February 2013

2 Abstract This dissertation discusses the development and implementation of a methodology for the buffeting response of cable-supported bridges, including uncertainty in the aeroelastic input (i.e., flutter derivatives, FDs). Flutter derivatives are the most important part of the loading and are estimated in a wind tunnel experiment. A second order polynomial model ( model curve ) for the flutter derivatives is proposed. The coefficients of this polynomial are random variables, whose probability distribution is conditional on the reduced wind speed. For computational reasons in subsequent analysis, however, this dependency is neglected and the probability of these random variables is treated as independent of the reduced wind speed. For analysis purposes the first- and second-order statistics are estimated from experiments, treating all the wind speed data as part of the same population. Wind tunnel experiments are conducted at Northeastern University and a section model of a truss-type bridge deck is used. The simplified polynomial model for the FDs, including the second order description of its variability, is employed in the derivation of the probability of the onset of flutter using Monte-Carlo (MC) simulations. The simplified stochastic model curves for FDs are used to estimate the buffeting bridge response. In the standard approach the result of the buffeting analysis is the value of the RMS dynamic response at a given wind speed. In the proposed probabilistic setting one ii

3 estimates the probability that a given threshold for the variance of the response is exceeded. This probability is later used, together with information on the probability of the wind velocity at a given site, to predict the expected value of the loss function due to the buffeting response of a 1200-meter suspension bridge (a function proportional to the cost associated with interventions needed to ensure safety). iii

4 Acknowledgments I would like to most sincerely thank my advisor Dr. Luca Caracoglia for his constant guidance, teaching, and support throughout this rather complicated and intense research. He has motivated and encouraged me towards improvement and excellence in research. I have been grateful to work with him and to be part of his research team. I would also like to thank my PhD committee members, Professor George G. Adams, Professor Dionisio P. Bernal and Professor Mehrdad Sasani for their overall encouragement through the PhD studies and for valuable comments and recommendations. The PhD studies, described by this research, were supported in part by the National Science Foundation of the United States (NSF), Award No from 2008 to Any opinions, findings, conclusions and recommendations are those of the writer and do not necessarily reflect the views of the NSF. The Department of Civil and Environmental Engineering is also acknowledged for providing support from 2010 until the completion of the program in the form of teaching assistantship. Finally, I am deeply grateful to my parents and younger brother, whose love and support has always been a tremendous source of strength and encouragement for me. They never lost faith in me and are always willing to provide a helping hand. Their love and invaluable support gave me the motivation to accomplish many goals in my life. iv

5 Table of Contents Abstract ii Acknowledgments... iv Table of Contents... v List of Tables... viii List of Figures... ix Nomenclature... xiii Chapter 1 1 Introduction Motivation Outline... 6 Chapter 2 9 Wind-Induced Response of Long-Span Bridges: Review General Formulation Background on Flutter Background on Buffeting Effect of Wind Directionality: Skew Wind Theory Chapter 3 31 A Second-order Polynomial Model for Flutter Derivatives Description of the Polynomial Model and Discussion on its Physical Interpretation Description of the Polynomial Model Discussion on the Selection of the Polynomial Model, based on Physical Behavior of Flutter Derivatives Description of the Wind Tunnel, used for Experimental Verification of the Polynomial Model Description of the Experimental setups, used for Verification Description of the Aeroelastic Section-Model, used for Verification v

6 3.5 Description of the Tests and Experimental Identification Reason for the Use of the Polynomial Model in the Context of Random Flutter Derivatives Estimation of Variance and Co-variance of Cj and Dj coefficients of the Model Cures from Experiments Summary of Experimental Results and Comparison with Literature Data ( Jain s Data ) Chapter 4 58 A Methodology for the Analysis of Long-Span Bridge Buffeting Response, accounting for Variability in Flutter Derivatives Introduction Multi-Mode Buffeting Analysis ( Deterministic Case ) Validation for Closed-Form Solution Monte-Carlo and Quasi-Monte-Carlo Methods Examination of the Computational Efficiency of the MC and QMC Methods for Calculating the Double Integral in Eq. (2.24) Monte-Carlo-based Methodology for Buffeting Analysis Considering Uncertainty in the Flutter Derivative ( Statistical Case ) Description of the Bridge Example and RMS Threshold Levels ( Probabilistic Setting ) TEP Curves using Literature Data TEP Curves using NEU s Flutter Derivative Data Effect of Wind Directionality on Statistical Buffeting Response: TEP Surfaces Exploratory Performance Analysis on a Full-Scale Structure Summary Chapter Lifetime Cost Analysis due to Buffeting Response on a Long-Span Bridge, accounting for Variability in Flutter Derivatives Introduction Peak Estimation via RMS Response Lifetime Expected Cost Analysis Description of the Structural and Aeroelastic Model vi

7 5.5 Estimation of Peak Dynamic Response during Buffeting Monte-Carlo-based Methodology for Statistical Buffeting Analysis considering Uncertainty in the FD Wind-Direction-Independent Analysis Wind-Direction-Dependent Analysis Flutter Analysis: Numerical Results Lifetime Expected Intervention Cost Analysis - Numerical Results Estimation of the Limit-State Probabilities Pj from TEP Analysis Expected Intervention Cost - Description of the Simulations Expected Intervention Cost - Numerical Results using NEU s FD Data Discussion and Remarks Chapter Summary and Conclusions Summary Conclusions Recommendations for Future Research Outcome of the PhD Studies: List of Deliverables Journal Publications (Published/under review) Other Journal Publications (not related to the main topic of this Dissertation) Full Papers in Conference Proceedings Other Papers Published as Conference Proceedings (not related to the main topic of the Dissertation) Poster Presentations References vii

8 List of Tables Table 3.1 The static coefficients and their derivatives at α0 (Jain et al., 1998) Table 4.1 Natural frequencies and mode types of Golden Gate Bridge (Jain 1996) Table 4.2 Comparison of closed-form numerical solution with literature results at l/ Table 4.3 Bias and relative errors in the MC case: (a) for heave σhh; (b) for torsion σαα Table 4.4 Bias and relative errors in the QMC case: (a) for heave σhh; (b) for torsion σαα.. 85 Table 4.5 Threshold values for σhh and σαα, employed in the TEP analysis with flutter derivatives from the literature Table 4.6 Study cases used for serviceability on full-scale structure Table 5.1 Structural performance thresholds for vertical deck response (Tj) Table 5.2 Probabilities of each damage state (Pj) due to buffeting response based on the structural performance thresholds (T = Tj) using NEU s FD data viii

9 List of Figures Figure 1.1 Tacoma Narrows Bridge collapsed in 1940 due to wind-induced torsional flutter (reproduced from Simiu and Scanlan, 1996) Figure 2.1 A suspension bridge and a section of the deck (Schematic view of a generic finiteelement model of the structure) Figure 3.1 NEU-MIE wind tunnel (Brito 2008) Figure 3.2 Experimental setup: (a) NEU s small-scale wind tunnel; (b) NEU s Aeroelastic Force Balance with the truss-type bridge deck model Figure 3.3 The Golden Gate Bridge (Photo courtesy of Google Image) Figure 3.4 Truss-type deck section model, replicated the features of the Golden Gate Bridge at a scale 1:360; model width is B = 76 mm and the aspect ratio is B/D = 3.5: Figure 3.5 Flutter derivatives of a truss-type section model with aspect ratio B/D= 3.5:1 measured at NEU: (a) H1 * ; (b) H2 * ; (c) H3 * ; (d) H4 * ; (e) A1 * ; (f) A2 * ; (g) A3 * ; (h) A4 * Figure 3.6 Flutter derivatives of a truss-type section model(the Golden Gate Birdge) derived from (Jain et al., 1998): (a) heave Hi * (i=1,,4); (b) torsion Ai * (i=1,,4) Figure 4.1 Flowchart describing the MC-based methodology for buffeting analysis Figure 4.2 Two-dimensional sample points = 1,000: (a) MC with uniform distribution, (b) QMC with Halton sequence Figure 4.3 Ten simplified (sinusoidal-like) mode shapes used in the multi-mode buffeting analysis: (a) LS, Hz; (b) VAS, 0.087Hz; (c) LAS, Hz; (d) VS, Hz; (e) VAS, Hz; (f) VS, Hz; (g) TAS, Hz; (h) TS, Hz; (i) VAS, Hz; (j) VS, Hz. (Note: L is Lateral, V is Vertical, T is Torsional, S is Symmetric, AS is Anti-symmetric) Figure 4.4 MC-based scatter plots of RMS response for deck section at x= l/4 and for U = 22.2 m/s as a function of NMC: (a) heave σhh; (b) torsion σαα Figure 4.5 Coefficient of variation of the RMS response at x = l/4 for U = 22.2 m/s, computed by MC algorithm: (a) heave σhh; (b) torsion σαα ix

10 Figure 4.6 Tolerance intervals for vertical RMS response (σhh) of 100 MC simulations: (a) NMC = 1,000; (b) NMC = 5,000; (c) NMC = 10,000; (d) NMC = 50,000; (e) NMC = 100, Figure 4.7 Tolerance intervals for torsional RMS response (σαα) of 100 MC simulations: (a) NMC = 1,000; (b) NMC = 5,000; (c) NMC = 10,000; (d) NMC = 50,000; (e) NMC = 100, Figure 4.8 QMC-based scatter plots of RMS response for deck section at x= l/4 and for U = 22.2 m/s as a function of NQMC: (a) heave σhh; (b) torsion σαα Figure 4.9 Coefficient of variation of the RMS response at x = l/4 for U = 22.2 m/s, computed by QMC algorithm: (a) heave σhh; (b) torsion σαα Figure 4.10 Tolerance intervals for vertical RMS response of 100 MC simulations (σhh): (a) NQMC = 1,000; (b) NQMC = 5,000; (a) NQMC = 10, Figure 4.11 Tolerance intervals for RMS response of 100 MC simulations (σαα): (a) NQMC = 1,000; (b) NQMC = 5,000; (a) NQMC = 10, Figure 4.12 RMS values for vertical buffeting response as a function of wind speed U corresponding to a given confidence level Figure 4.13 Flutter derivatives H1 * (a) and A2 * (b) of the Golden-Gate Bridge girder with aspect ratio B/D = 3.5:1. Data sets were reproduced from (Jain 1996; Jain et al. 1996) with α0=0. The ( reference ) coefficients of the Polynomial Model were derived by regression of the data sets, according to Eqs. (3.1) and (3.2). Tolerance limits (dashed lines) were based on approximate evaluation of one standard deviation Figure 4.14 TEP curves of RMS response with respect to thresholds T1 to T3 at the deck section l/4: (a) σhh; (b) σαα (DFV: Deterministic Flutter Velocity) Figure 4.15 Procedure for rescaling the TEP curves in Fig. 6.11(c) based on Eq. (6.3): (a) prior probability or TEP; (b) marginal likelihood function; (c) Posterior probability or TEP (DFV: Deterministic Flutter Velocity) Figure 4.16 TEP curves of RMS response at deck section l/4 (T2 threshold only) before (T2) and after rescaling (T2M): (a) σhh; (b) σαα (DFV: Deterministic Flutter Velocity) Figure 4.17 TEP curves of RMS responses with thresholds based on the RMS displacement, deck section at l/4 and NEU s flutter derivatives: (a) σhh; (b) σαα Figure 4.18 TEP surfaces of RMS displacement for T2M threshold as a function of wind accounting for effects of skew winds at l/4 with literature flutter derivatives: (a) σhh at l/4; (b) σαα at l/4; (c) σhh at l/2; (d) σαα at l/ x

11 Figure 4.19 TEP surfaces of RMS displacement for T2 threshold as a function of wind accounting for effects of skew winds at l/4 with NEU s flutter derivatives: (a) σhh at l/4; (b) σαα at l/4; (c) σhh at l/2; (d) σαα at l/ Figure 4.20 National Data Buoy Center (NOAA Station , Latitude: N, Longitude: W) (Photo reproduced from NOAA, Figure 4.21 PDFs of parent (continuous time) mean wind velocity and annual maxima of mean wind velocity, data from NOAA (NOAA) Figure 5.1 Reference peak vertical dynamic response, ( deterministic without variability in FD) as a function of wind velocity at θ = 0 with both Jain s flutter derivatives and NEU s flutter derivatives at l/4: (a) displacement; (b) acceleration Figure 5.2 TEP curves of the peak dynamic response with respect to thresholds T1, T2, T3 deck section at l/4 using Jain s FD data: (a) vertical response; (b) torsional response Figure 5.3 Recaled TEP curves (modified by Eq. 4.3) of the peak dynamic response with respect to thresholds T1, T2, T3 deck section at l/4 using Jain s FD data: (a) vertical response; (b) torsional response Figure 5.4 Rescale TEP curves (modified by Eq. 4.3) of the peak dynamic response with respect to thresholds T1, T2, T3 deck section at l/4 using NEU s FD data: (a) vertical response; (b) torsional esponse Figure 5.5 A comparison between two sets of curves for vertical response based on NEU s FD data (continuous lines) and Jain s FD data (dotted lines): (a) T1=20 milli-g; (b) T2=1m; (c) T3=2m Figure 5.6 A comparison between two sets of curves for torsional response based on NEU s FD data (continuous lines) and Jain s FD data (dotted lines): (a) T1=20 milli-g; (b) T2=1m; (c) T3=2m Figure 5.7 Recaled TEP surfaces (modified by Eq. 4.3) of the peak dynamic vertical response with various intervention levels at l/4 using Jain s FD data: (a) T1=20 milli-g; (b) T2=1m; (c) T3=2m Figure 5.8 Recaled TEP surfaces (modified by Eq. 4.3) of the peak dynamic (normalized to B/2) torsional response with various intervention levels at l/4 using Jain s FD data: (a) T1=20 milli-g; (b) T2=1m; (c) T3=2m Figure 5.9 Recaled TEP surfaces (modified by Eq. 4.3) of the peak dynamic vertical response with various intervention levels at l/4 using NEU s FD data: (a) T1=20 milli-g; (b) T2=1m; (c) T3=2m xi

12 Figure 5.10 Recaled TEP surfaces (modified by Eq. 4.3) of the peak dynamic (normalized to B/2) torsional response with various intervention levels at l/4 using NEU s FD data: (a) T1=20 milli-g; (b) T2=1m; (c) T3=2m Figure 5.11 Probability distributions of flutter velocity, Ucr, using NEU s flutter derivatives: a) PDF (probability density function); b) CDF (cumulative density function). 154 Figure 5.12 Resolution of the Monte-Carlo-based flutter procedure vs. standard error Figure 5.13 Intervention costs normalized to the initial construction cost for user comfort level threshold T1=20 milli-g over time based on NEU s FD data CE EC t C0 C0 : (a) 3D PMF (probability mass function stem plot); (b) 2D expected normalized cost, - discount rate/year λ= Figure 5.14 Expected intervention costs normalized to the initial construction cost due to deformation in deck based on NEU s FD data CE EC t C0 C0 : (a) lower tolerance case (T2=1m); (b) higher tolerance case (T3=2m), - discount rate/year λ= xii

13 Nomenclature The following symbols were used in this dissertation: D * * A damping matrix of the system A,.., A flutter derivatives per unit length, torsional moment b,..., b constant parameters in Eqs. (5.11a) to (11e) B stiffness matrix of the system B bridge deck width c decay factor 0 C, C random parameters in Eq. (4.1) ( j = 1,3,5,7) i C t C i1 C j total cost of the structure at time t (years) initial cost of the structure cost in present dollar value C, C, C drag, lift, moment static coefficients D L M CM, E expected value of maintenance and repair cost dm infinitesimal inertia d search direction at step n (FORM) n i1 1, ref 4, ref B/ D bridge deck, aspect ratio D bridge deck height D, D random parameters in Eq. (4.2) ( j = 1,3,5,7) i,..., D "reference" parameters in Eqs. (5.14) (average solution with no errors) ' ' D1,..., D4 mean-removed random C1,..., C4 E[] expectation operator Ε system matrix for two-mode aeroeastic instability analyses * Ε complex conjugate transpose of matrix E F loss of performance for the bridge v U, U marginal probability density function of U marginal probability density function of g( U, U ) limit state function cr f f f site t1, t1 v1, v1 joint probability density function of, U g gust effect factor G,..., G modal integrals, simulated bridges xiii

14 hxt (, ) vertical bridge oscillation, simulated bridges h ( x), h ( x) modal eigenfunctions, vertical oscillation ( g-th mode and mode v1) g v1 h dimensionless eigen-function of i-th mode shape i * * 1 4 h ˆ h * i 0 dh/ dt peak vertical acceleration H,.., H flutter derivatives per unit length, lift force H vertical flutter derivatives ( i = 1,...,4) iˆ imaginary unit I identity matrix I g t1 v1 mass moment of inertia per unit length of the deck, simulated bridges I indicator function I, I generalized modal inertias (modes i, j) i j I, I generalized modal inertia for modes t1 and v1 K, K reduced frequency, reduced critical frequency cr K, K reduced modal frequencies for modes t1 and v1 t1 v1 K n l 0 effective reduced frequency central span length of the simulated bridges m mass per unit length of the bridge deck, simulated bridges D, L, M aeroelastic drag, lift, moment forces per unit length (Fig. 2.2) ae ae ae D, L, M bufetting drag, lift, moment forces per unit length (Fig. 2.2) b b b n, n, n still-air natural frequency (any, torsional mode t1, vertical mode v1), Hz t1 v1 N N MC number of wind tunnel data points number of Monte Carlo points pxt (, ) lateral vibration of the simulated bridges p dimensionless eigen-function of the i-th mode i P[] probability function P direction-dependent flutter probabilities F P loss of performance of the bridge (threshold T) P T P probabilities of each damage state j j U cr P f q i q, q dimensionless modal groups (e.g., q 0.5 B L/ I ) 4 t1 v1 t1 t1 Q b i flutter probability flutter probability i-mode shape of the bridge generalized force vector Q generalized force of the i-th mode xiv

15 E standard Gaussian vector standard error S x, x, K cross PSD of modal loading between two generic sections x and x F A B A B S, S, S auto and cross PSD function of the dynamic response hh h S S S QQ i j S S uu ww T t uxt (, ) lateral component of turbulent wind U u u * modal force cross-spectra Kaimal spectrum Lumley-Panofsky spectrum generic threshold time velocity variable used in conditional probability functions (with u U) friction velocity u* design point U U U U cr n p Rcr, wind speed, m/s critical torsional-flutter speed, m/s normal component of wind speed to the deck, m/s parallel component of wind speed to the deck, m/s U reduced velocity ( U /( nb)) U R site 0 reduced critical velocity (at flutter); real root of Eq. (5.12) wind speed at the bridge site, m/s vx (, t) horizontal component of turbulent wind wxt (, ) vertical component of turbulent wind x z longitudinal coordinate along the bridge axis terrain roughness length ( xt, ) torsional vibration of the simulated bridges t1 i 0 ( x), ( x) mode-shape functions, torsional oscillation of the simulated bridges g αˆ unit-gradient row vector (FORM) n ij dimensionless eigen-function mean-wind attack angle of the deck generalized safety index Kronecker delta function tolerance parameter (FORM) dimensionless variance reduction coefficient xv

16 t1 v1 t1 v1 t1 v1, structural modal damping with respect to critical, modes t1 and v1 i generalized coordinate ( t), ( t) generalized modal coordinate, modes t1 and v1 ( K), ( K) Fourier transform of the generalized modal coordinate, modes t1 and v1 ti, T v1 t1 ξ two-mode flutter eigenvector =,, with transpose operator n amplitude parameter in FORM single-mode torsional instability equation (see Eq. 5.9) t tcr, torsional-mode dimensionless ratio ( K / K) value of at critical velocity air density t, correlation coefficients of random parameters D, D D3, D4 D5, D6 3 4 ' '' i, i maximum positive and minimum negative differences t1 v1 standard Gaussian comulative density function frequency ratio, simple harmonic motion t1 and D, D 5 6,, still-air circular frequency (any, torsional and heaving mode), rad/s cr,, vertical, torsional and lateral RMS displacements hh critical-flutter circular frequency (torsional mode), rad/s pp Pf variance of the flutter-probability MC estimator still-air circular frequency, rad/s i i-th mode natural circular frequency, rad/s T Subscripts and superscripts: g n generic mode index for the simulated bridges generic step of iteration (FORM) t1, v1 fundamental bridge modes, torsional and vertical Moore-Penrose pseudo-inverse xvi

17 Chapter 1 Introduction The quantitative analysis of aerodynamic effects on long-span bridges has been considered (Simiu and Scanlan 1996), since the collapse of the Tacoma Narrows Bridge in 1940 (shown in Fig. 1.1). Wind engineering researchers have devoted great efforts to understand windinduced or aeroelastic phenomena (Davenport 1962; Scanlan and Tomko 1971), associated with the vibration of long-span bridges. Aeroelastic is used to indicate fluid-structure interaction between a flexible structural system and wind air flow. Investigations have also been performed by many researchers (Kwon 2010; Namini et al. 1992; Scanlan 1987; 1993; Scanlan and Jones 1990b) to prevent these loading mechanisms from adversely affecting the satisfactory performance of long-span bridges. The accurate assessment of fluctuating wind loads on long-span bridges is necessary to avoid failures or undesired vibrations. Two types of aeroelastic phenomena, namely flutter and buffeting response, are considered in this study. There are the two phenomena that are relevant for the analysis of bridge deck. Both the potential collapse of the structure 1

18 2 due to flutter instability and the dynamic vibration due to wind turbulence (buffeting) at moderate to high wind speeds are important for bridge design. Flutter is defined as an oscillatory instability, induced in the bridge deck, when a bridge is exposed to a wind speed above a certain critical threshold. Beyond this limit, diverging vibration of the deck is possible, which may result in a catastrophic structural failure. A classic example of such failures is illustrated in Fig 1.1. Instability can be predicted through the analysis of flutter derivatives (FDs), and needless to say must be avoided by all means. Buffeting is defined as the dynamic vibration regime due to fluctuating loading, promoted by wind turbulence, which is also influenced by the interaction with structural deck motion. The bridge vibration is stochastic due to oncoming-flow turbulence and signature turbulence, produced around the deck girder through flow separation and air recirculation (e.g., (Jones and Scanlan 2001)). The dynamic amplification of vibration, which causes buffeting, is often observed on long-span bridges (Miyata et al. 2002; Xu et al. 2007; Xu and Zhu 2005). Buffeting does not usually lead to catastrophic failure of the bridge. However, vibrations cannot be avoided but need to be monitored since they can affect the serviceability; in fact, damage, fatigue in selected structural elements and user discomfort are possible. Both phenomena may occur either separately or together, and can be predicted by utilizing experimentation in wind tunnel (Bienkiewicz 1987; Ehsan et al. 1993; Huston et al. 1988). Such experimentation is essential for the derivation of wind-induced forces, especially the loading terms on the deck triggered by fluid-structure interaction; these can

19 3 be expressed in terms of flutter derivatives, originally developed by (Scanlan and Tomko 1971), which are non-dimensional aerodynamic force coefficients per unit deck length as a function of the reduced velocity. Flutter derivatives are the essential parameters in the estimations of the critical wind velocity of flutter instability and the buffeting response of long-span cable supported bridges. Recently, it has been demonstrated that experimentally-derived FDs are random in nature with uncertainty affected by measurement errors (Sarkar et al. 2009); this uncertainty in the measurement of the FDs are unavoidable during testing in the wind tunnel. To assess such uncertainties and the effects on both flutter and buffeting response, it is necessary to develop specific analysis tools which could enable accurate bridge performance assessment. This dissertation proposes to develop a methodology for deriving the solution to buffeting problem on long-span bridges, which will involve the direct representation of the above-described sources of uncertainty in the aeroelastic input through appropriate statistical analysis of the FDs. A second order polynomial model for the FD is proposed and labeled as model curve in this study. The coefficients of this polynomial are treated as random variables, whose probability distribution is conditional on the reduced wind speed. For computational reasons in subsequent analysis, however, this dependency is neglected and the probability of these random variables is treated as independent of the reduced wind speed. For analysis purposes the first- and second-order statistics are estimated from experiments, treating all the wind speed data as part of the same population. Wind tunnel experiments are used to validate the

20 4 proposed methodology and to confirm the relevance of measurement errors in these aeroelastic force terms. Wind tunnel tests have been conducted at Northeastern University (NEU) for this purpose. 1.1 Motivation The prediction and simulation of long-span cable-supported bridge dynamic response due to wind hazards are particularly difficult in consideration of the inherent complexity of the wind field, turbulence fluctuations and pressure distributions around the deck (which is the most vulnerable part of the structure). Experimentation is essential for the derivation of this dynamic loading. Despite the efforts of the research community towards the development of refined techniques to simulate full-scale bridge response (Ozkan 2003; Ozkan and Jones 2003), observations can differ from the simulations of the response, based on wind-tunnel experiments and measurements of equivalent loading. These discrepancies are associated with the experimental procedures (and their errors) in wind engineering and must be carefully accounted for in the existing simulation methods. Reliability analysis against flutter, considering uncertainty in the FDs, has been investigated by a few researchers (Dragomirescu et al. 2003; Ge et al. 2000; Kwon 2010; Mannini and Bartoli 2007; Ostenfeld-Rosenthal et al. 1992; Pourzeynali and Datta 2002; Scanlan 1999). Recently, it has also been shown how to experimentally estimate the statistical moments of the FDs (variance, co-variance, etc.) from data extracted in wind tunnel tests (Kwon 2010; Mannini et al. 2012).

21 5 However, very limited emphasis has been given to structural serviceability due to uncertainty in buffeting loading (Caracoglia 2008a). With the aging bridge inventory in the United States, it is therefore important to advance the current analysis approaches to include the statistical buffeting response. The term statistical buffeting analysis is coined for the first time in this dissertation to differentiate from the standard buffeting analysis in the absence of uncertainty in the FD. This dissertation focuses on the development of a methodology for statistical buffeting analysis, including the uncertainty in the FD. To accomplish these tasks and, most importantly, a second order polynomial model ( model curve ) for the FD is proposed. The model curve is a second order polynomial description of the FDs where uncertainty is associated with coefficient of the polynomial. This curve is used to describe the behavior of the flutter derivatives as a function of reduced velocity. The physical justification for this selection stems from the observation that the FDs tend to follow a general trend, especially for moderately bluff deck sections (Simiu and Scanlan 1996). Therefore, postulating such a model curve for FDs was selected as an appropriate assumption for projecting the variability of the coefficients of the model curve into the analysis of the buffeting response of the bridge. In the standard approach the result of the buffeting analysis is the value of the RMS dynamic response at a given wind speed. In the proposed probabilistic setting one estimates the probability that a given threshold for the variance of the response is exceeded. This probability is later used, together with information on the probability of the wind velocity at a given site, to predict lifetime expected cost (Wen and Kang 2001) due to the buffeting

22 6 response of a 1200-meter suspension bridge (a function proportional to the cost associated with interventions needed to ensure safety or for maintenance) is affected by the variability of the FDs. Even though the ultimate goal of the research is the development of a generalized methodology for the solution to buffeting problems on long-span bridges for the analysis of the effects induced by various sources of uncertainty; this should involve the extension of the procedures and should include both wind-loading input and selected structural properties. This dissertation represents a first step towards this objective. 1.2 Outline This dissertation is divided into the following chapters, after providing a general introduction and motivation in Chapter 1. Chapter 2 summarizes the background theory of wind-induced response of long-span bridges and reviews the fundamental aspects of aerodynamics and aeroelasticity of longspan suspended bridge decks. Chapter 3 describes the development of a model curve for representation of the behavior of flutter derivatives as a function of reduced wind velocity. This chapter also describes the experimental setup, measurements and experimental results, used in this research. Flutter derivatives were measured in the wind tunnel at NEU. Chapter 4 describes the standard buffeting analysis of long-span bridges ( deterministic case ) as well as statistical buffeting analysis which includes the variability in the FD. The statistical buffeting response was evaluated by adopting the concept of

23 7 fragility ; this was employed in the calculation of the exceedance probability of preselected vibration thresholds, conditional on mean wind speed and direction at the deck level. Chapter 5 discusses the lifetime estimation of monetary losses for a long-span bridge, designated as cost analysis, due to buffeting response. The expected value of the loss function (lifetime cost estimation) of a 1200-meter suspension bridge is evaluated by applying the statistical buffeting analysis. Summary of the work, conclusions of the dissertation and directions for future research are discussed in Chapter 6.

24 8 Figure 1.1 Tacoma Narrows Bridge collapsed in 1940 due to wind-induced torsional flutter (reproduced from Simiu and Scanlan, 1996).

25 Chapter 2 Wind-Induced Response of Long-Span Bridges: Review This chapter reviews the fundamental aspects of aerodynamics and aeroelasticity of longspan suspended bridge decks. Figures 2.1 and 2.2 show a section view of the bridge deck to be analyzed. The wind-induced dynamic response of a long-span bridge close to aeroelastic instability is most conveniently analyzed in the frequency domain, as described by (Jain 1996; Jones and Scanlan 2001; Katsuchi et al. 1999); this method, referred to as multimode approach in wind engineering, is reviewed in this chapter. 2.1 General Formulation The deflection components of the bridge deck (i.e., h(x,t), p(x,t) and α(x,t) in Fig. 2.2) can be expressed in terms of the generalized coordinate of the mode ξi(t), the deck width B and the dimensionless representations of the i-th mode form along the deck hi(x), pi(x) and αi(x) as 9

26 10 vertical : hxt (, ) h x B t, (2.1a) i i i (2.1b) torsional : (x,t) i x i t. (2.1c) i lateral : p( xt, ) p x B t, i i i In Eq. (2.1), hi(x), pi(x) and αi(x) are dimensionless eigen-functions associated with the i-th mode shapes of the deck, ξi(t) are generalized coordinates. x is the coordinate along the deck span and t is time (Fig. 2.2). The linear dynamic response of the bridge deck is derived by standard modal expansion of the vibration in terms deck modes, as indicated in Eqs. (2.1); the original formulation, described for example by Jain (1996), is based on the representation of the deck girder as a continuous (i.e., beam-type) element oriented along the x axis, transversely rigid section by section, and with lateral and vertical vibration with respect to the centroid C of the deck section (Fig. 2.2) in the directions orthogonal to x (h and p) and torsional rotation about x. Only vertical, lateral, torsional components of the deck section are used herein because these three are considered as primary deflection components due to wind loadings for long-span suspension bridges. A review of modal expansion techniques for structural dynamic analysis of continuum systems is not included in this chapter but may be found, for example, in (Meirovitch 1970). It must be noted that normalization of the eigen-functions in Eq. (2.1) is performed with respect to a reference dimension B, coincident with the deck width. This normalization was first introduced by (Scanlan and Jones 1990a) to enable the subsequent derivation of the dynamic modal response equation in a general form, regardless of the specific features of the generic mode i (lateral, vertical, torsional, etc.).

27 11 The governing generalized equation of motion of mode i (ξi) therefore becomes (e.g., Scanlan and Jones, 1990) I i i 2 i i i i2 i Q t i, (2.2) where Ii and Qi(t) are the generalized inertia and modal force of the i-th mode, ωi and ζi are the i-th mode natural circular frequency and the modal damping ratio. The generalized inertia Ii is defined as I q x, y, zdmxy,, z, (2.3a) i structure 2 i where qi (x, y, z) are the i-th mode shape of the bridge and dm is an infinitesimal inertia; this equation is written in a general form to emphasize that the integration may be carried out over the entire structure to also account for modal mass contributions from portions of the bridge other than the deck itself, which are also involved in the vibration; these terms depend, for example, on the moving cables in a suspension bridge or the tower motion. If the mass and moment of inertia of the moving deck and cables are assumed as constant along x, and respectively expressed by m0 and I0 per unit deck length, the modal inertia, using the expansion in Eq. (2.3a), simply becomes l i 0 0 i 0 i 0 i I m h x B m p x B I x dx, (2.3b)

28 12 in which l is the total deck length. (2.4). The generalized force Qi(t) due to wind loading is given in a similar form by Eq. l Q t L x, t h x B D x, t p x B M x, t x dx, (2.4) i 0 i i i where L(x,t), D(x,t) and M(x,t) represent the lift, drag and pitching moment per unit span length (in Fig. 2.2). They are assumed to be separable into motion-dependent loads and turbulence-induced loads (motion-independent) and are defined as lift : L Lae Lb (2.5a) drag: DDae Db (2.5b) moment : M Mae Mb (2.5c) where the subscripts ae and b refer to aeroelastic and buffeting loads, respectively. The aeroelastic (or self-excited forces) are assumed to be linearizable. For purely sinusoidal motions of frequency ω, the aeroelastic forces can be expressed as (Scanlan and Tomko 1971). L ae 1 2 U 2 * h B KH 1 U KH * B 2 U K 2 H * 3 K 2 * h H 4 B KH * p 5 U K 2 * p H 6 B, (2.6a)

29 13 D ae 1 2 U 2 * p B KP 1 U KP * B 2 U K 2 P * 3 K 2 * p P 4 B KP * h 5 U K 2 * h P 6 B, M ae 1 2 U 2 B 2 * h KA 1 U KA * B 2 U K 2 A * 3 K 2 * h A 4 B KA * p 5 U K 2 * p A 6 B, (2.6b) (2.6c) where is the air density, U the mean velocity of the oncoming wind (which is turbulent in general) at the deck level, K (= ωb/u) is the reduced frequency with ω circular frequency; h dh / dt, d / dt and p dp / dt pertain to the deck section at x. In Eq. (2.6), Hi * Pi * and Ai * (with i = 1,,6) are flutter derivatives of the deck cross-section. As noted by Scanlan and Tomko (1971), the previous expressions are written in a mixed time-frequency form and are valid for simple harmonic motion of the deck at a given ω; this assumption is strictly valid at flutter but is acceptable in the case of vibration induced by turbulence disturbances on low-damping systems, such a suspension bridge. Equations were derived by extension of the standard airfoil theory (Theodorsen 1935) to non-aerodynamic (bluff) bridge deck sections. It must be noted that the frequency-time duality is only apparent since it disappears once Fourier-domain analysis is employed to derive the bridge response (Scanlan and Tomko 1971), as later described. Buffeting forces per unit length are fluctuating loads which can be described by quasi-steady theory and turbulence disturbances for vibration about a static equilibrium configuration of the deck due to mean wind loads. For mean incident wind orthogonal to the bridge axis, the loading depends on the vertical (w), and lateral (u) turbulence, which are stochastic quantities as a function of time t and position x along the deck (Fig. 2.2). For small vibration amplitudes these can be obtained by first-order expansion about on equilibrium

30 14 position under mean wind, described by an angle of attack α0 in the vertical plane (Fig. 2.2). As an example, the lift force Lb, drag force Db and moment Mb per unit deck length are defined as (Jones and Scanlan 2001) dc w x t 1 2 u x, t, L Lbx, t U BCL2 CD, 2 U d U dc w x t 1 2 u x, t, D Dbx, t U BCD 2, 2 U d U dc w x t u x, t, M Mbx, t U B CM 2, 2 U d U (2.7a) (2.7b) (2.7c) with CL, CD and CM being lift, drag and moment static coefficients (referred to deck width B) of a typical deck section, evaluated at mean-wind attack angle of the deck α0; u(x,t), w(x,t) are the along-wind and vertical components of turbulent wind. Span-wise correlation loss of the turbulence-induced forces along the x direction (from Eq. 2.7) was incorporated into the formulation to calculate the generalized loading Qi. The standard exponentially decreasing coherence model was employed (e.g., Jones and Scanlan, 2001). This model is later described in Section 2.3. For example, for single-mode analysis, Eq. (2.1) can be further simplified as hxt (, ) hxb ( ) ( t), (2.8a) p( xt, ) pxb ( ) ( t), (2.8b) ( x, t) h( x) ( t). (2.8c)

31 15 Similarly, the dynamic loading Qi in Eq. (2.4) becomes a scalar term. The multi-mode system of equations can be formed by separating the generalized loading Qi(t) into aeroelastic and buffeting components Qi(t)=Qae,i(t)+Qb,i(t) and by recognizing that the loading induced by Qae,i(t) is linearly dependent quantities related to dynamic motion and velocity of the deck sections; therefore, Qae,i(t) can be expressed as a linear function of the generalized displacements and velocities through Eq. (2.1) and the effect of Qae,i(t) on the bridge can be interpreted as a modification to the generalized stiffness and damping of the structural modes which depend on wind speed U. A more detailed description may be found in Scanlan and Jones (1990). The generic scalar Eq. (2.2) can be simplified as i 2 i i i i2 i Q t ae,i I i Q b,i t I i and the left-hand side rewritten by regrouping terms as a function of Qae,i(t)/Ii as wind-induced stiffness and damping equivalent quantities. In matrix notation the dynamic system becomes, after modal truncation to a significant set of modes, s I '' A ' BQ b, (2.9) where ξ is the generalized coordinate vector, ( ) represents a derivative with respect to a dimensionless time (Jain et al. 1996) s = Ut/B, I is the identity matrix, A and B are the damping and stiffness matrices of the system, which are no longer diagonal since mode coupling is induced by fluid-structure interaction via aeroelastic loads; finally, Qb is the generalized buffeting force vector.

32 16 The general terms of A, B and Qb can be expressed as (Jones and Scanlan 2001) 4 BlK * * * * ij K ikiij HG 1 hih HG j 2 hi HG j 5 hip PG j 1 pip j 2Ii A 2 [ PG PG AG AG AG * * * * * 2 pij 5 ph i j 1 ihj 2 i j 5 ipj BlK B [ 4 2 * * * * ij KKi ij HG 3 hi HG j 4 hih HG j 6 hip PG j 1 pi j 2Ii PG PG AG AG AG * * * * * 4 pp i j 6 ph i j 3 i j 4 ihj 6 ipj ], ], (2.10) (2.11) Q {L, D, M, } dx l, (2.12) 4 B l l b s i 2 0 b x s hi b x s pi b x s i Ii where Ki=Bωi/U and the dependency on K=Bω/U is due to the flutter derivatives; δij is the Kronecker delta function defined as 1 ij 0 i i j. j (2.13) Note that the diagonal terms (i = j) in Eqs. (2.10) and (2.11) represent the single degree of freedom (and uncoupled) equations. The off-diagonal terms permit the aeroelastic coupling through the flutter derivatives and through the cross-modal integrals among different modes. The modal integrals between the ri normalized displacement or rotation of mode i and the corresponding sj component of mode j are denoted by G and are obtained by rs i j

33 17 G l dx r() x s () x, (2.14) l rs i j 0 i j where ri = hi, pi or αi, sj = hj, pj or αj. The new system of equations can be Fourier transformed into the reduced frequency (K) domain (Scanlan and Jones 1990a); for example a generic time-dependent function f(s) with s = Ut/B becomes in the frequency domain with iˆ 1: iks ˆ. (2.15) 0 f K f s e ds Consequently, a system of equations, exclusively dependent on K and, is derived from Eq. (2.9) as E Q (2.16) b, where and Q iks ˆ b are the Fourier-transformed K se ds and b 0 respectively. A general term of the impedance matrix E is Q vectors, 2 E K ika ˆ K B K. (2.17) ij ij ij ij

34 Background on Flutter A review of flutter theory is presented in this section. Information was derived from (Jain 1996; Katsuchi 1997). Flutter is an oscillatory instability induced when a bridge is exposed to a wind speed above a certain critical threshold. Beyond this limit, diverging vibration of the deck is possible, which may result in a catastrophic structural failure. One particular example was the Tacoma Narrows incident in 1940 when such phenomenon was clearly recognized. Flutter instability must be avoided by all means in bridge engineering. Aeroelastic instability can be predicted by analyzing the aeroelastic coefficients of bridge decks (flutter derivatives in Eq. 2.6) developed by (Scanlan and Tomko 1971), which are employed for simulating the dynamic response of the bridge. Flutter derivatives in Eq. (2.6) are force coefficients per unit length, routinely measured in wind tunnel tests. As briefly outlined in a previous section, these expressions of the shape-dependent force coefficients of the moving deck section have been strongly influenced by airfoil theory for streamlined bodies (Theodorsen 1935), from which they have been derived for use in civil engineering applications. Frequency-domain analysis was used in this study; this is the preferred method of bridge dynamic researchers because this analysis can be related to direct physical interpretation through flutter derivatives, obtained experimentally, as opposed to timedomain analysis, which requires modeling of the Qae,i(t) loading in terms of convolution integrals (Scanlan et al. 1974) and a more complex formulation (on occasion suffering from the lack of physical interpretation; (Caracoglia and Jones 2003).

35 19 The flutter condition is identified by solving the aeroelastically influenced effective damping problem derived from Eq. (2.16) by setting the turbulence-induced loading Q 0 (Jones and Scanlan 2001; Katsuchi et al. 1998). b E 0. (2.18) Equation (2.18) can be reduced to the nontrivial solution of the complex algebraic system. However, a direct method for the solution of det[e] is not available because the matrix E includes two unknown variables, K and ω. The matrix E also consists of complex numbers so that the condition of det[e]=0 must be satisfied with both the real and imaginary parts of the determinant simultaneously equal to zero. An iterative procedure is needed to solve for det[e]=0 (Jones and Scanlan 2001). This can be accomplished by fixing a value of K and seeking a value of ω, in the frequency range of interest, for which the determinant is zero, and changing the value of K until both the real and imaginary determinants are zeros at the same ω. Once the values K (= ωb/u) and ω are obtained with satisfying Eq. (2.18), the flutter speed can be calculated. For a multi-mode problem, the same procedure is required and the largest value of K of all solutions gives the flutter-critical condition. The mode corresponding to the solution of ω is the leading mode in the flutter condition. Moreover, the eigenvector ξ at the flutter condition gives the flutter mode shape which indicates the relative participation of each structural mode in flutter vibration.

36 Background on Buffeting A review of buffeting theory is presented in this section; the material has been derived from (Jain 1996; Katsuchi 1997). Buffeting is a dynamic phenomenon, in which the wind-induced loading is dynamic and due to wind turbulence. The bridge vibration is stochastic due to oncoming-flow turbulence or signature turbulence, enhanced by flow separation and recirculation around the deck girder (e.g., (Jones and Scanlan 2001)). The vector of buffeting forces on the right hand side of Eq. (2.16) is Q 0 F b1 l 4 Bl F 0 b2 I b 2 l 2 1 I1 1 1 In l l F 0 b n dx l dx, dx l (2.19) where the integrands in the vector above are the Fourier transforms of the Eq. (2.12). x K x Kh x x K p x x K x F, L, D, M,. (2.20) bi b i b i b i By substituting the terms L b, D b and M b from Eqs. (2.7a 2.7c) at the span location xa, Eq. (2.20) leads to

37 21 1 F b x, [{2 ( ) 2 ( ) 2 ( )} ( ) i A K CLhi xa CDpi xa CMi xa u K U C C h x C p x C x w K ' ' ' {( L D) i( A) 2 D i( A) M i( A)} ( )], (2.21) where notations CL =dcl/dα, CD =dcd/dα and CM =dcm/dα are used with the derivatives of the static coefficients with respect to the angle of attack evaluated at α0. Taking the complex conjugate transpose of the j-th equation at the location xb results in 1 x K C h x C p x C x u K U C C h x C p x C x w K * F b, [{2 ( ) 2 ( ) 2 ( )} ( ) i B L j B D j B M j B ' ' ' {( L D) j( B) 2 D j( B) M j( B)} ( )], (2.22) where the ( ) * represents the complex-conjugate-transpose operation. From Eq. (2.19), using Eqs. (2.21) and (2.22) the following matrix equation can be obtained (Jones and Scanlan 2001) QQ b * b 1 l l * dx 1 l l A dxb * dxa dxb FF 0 0 b1 b FF b1 bn 4 2 II 1 1 l l II 1 n l l Bl, 2U 1 l l * dx 1 l l A dxb * dxa dxb FF FF 0 0 bn b1 0 0 bn b n II n 1 l l II n n l l (2.23)

38 22 from which the turbulence-induced dynamic response of the bridge can be derived by standard random vibration techniques (e.g., (Newland 1993). The power spectral density (PSD) matrix of the generalized loading can be calculated, a general term of which is 4 2 Bl 1 QQ bi b j i A j B uu A B U IiI j l l,, S K q x q x S x x K dxa dxb rixarjx BSwwxA, xb, K, l l (2.24) where q i x 2 CLhi x CDpi x CM i x, (2.25) ' ' ' r x C C h x 2 C p x C x. (2.26) i L D i D j M j The equations above depend on the cross-power spectral densities of the lateral turbulence Suu(xA,xB,K) and vertical turbulence Sww(xA,xB,K). The uw-cross spectrum of the wind Suw has not been included in the equation above as it is usually of secondary importance on the dynamic response. A more complete formulation may be found in (Jain 1996). If the auto-power spectral density of the wind components is independent of the location x along the deck axis, the span-wise cross-spectral densities of the wind components in conventional form as (Scanlan and Tomko 1971)

39 23 ck FxA, xb, KFKexp xaxb, 2B (2.27) where c is a decay factor, the range of which is generally taken as 8nl 16nl c. (2.28a) U U simplified as Therefore, the cross-power spectral densities Suu(xA,xB,K) and Sww(xA,xB,K) can be ck Suu( xa, xb, K) S Kexp A B, uu x x 2 B ck Sww( xa, xb, K) S Kexp xa xb. ww 2 B (2.28b) The limits in Eq. (2.28b) can be used for force calculations to reflect the higher spanwise correlation in the pressure loading than the one seen in the velocity components (Larose 1992). Using the following expressions l l ck dxa dxb Hrs K r exp, i j 0 0 i xa sj xb xa xb 2B l l (2.29) where ri and si = hi, pi or αi, the ij-th term of the buffeting force matrix can be expressed as

40 Bl 1 QbQ i b ij uu ij ww j 2U IiIj Suu Sww S K Y K S K Y K S K, (2.30) where Suu ij L hih j D pip j M i j Y K C H C H C H Y Suu ij Y Suu ij K4CC L DHh 4 ip H j ph CC i j L M Hh i H j ihj K4 CC D M Hp H, i j ip j ' ' ' Sww ij L D h h D p p M Y K C C H C H C H i j i j i j ' ' ' ' CL CDCDHh ip H j ph C i j L CD CM Hh i H j ihj ' ' CC D M Hp. i H j ipj (2.31) (2.32) The power spectral of the wind components u and w in the atmospheric boundary layer, expressed as functions of K, are assumed as (Simiu and Scanlan 1996) S S uu ww K K 2 200zu* Kz U B zu* Kz U B 5/3 5/3,. (2.33) (2.34)

41 25 The first equation above is the Kaimal spectrum ; the second expansion is the Lumley-Panofsky spectrum. In the previous equations z is the elevation above ground and u* is the friction velocity, a function of the surface roughness. The friction velocity u* can be determined by using 1 z ln, k z (2.35) u* U z o where U(z) is the mean wind velocity at elevation z (usually taken as the velocity at the deck level, U), k is the von Kármán constant which is generally assumed to be k 0.4 (Simiu and Scanlan 1996) and z0 is the terrain roughness length. The power spectral density matrix for the generalized displacements ξ is developed in dimensionless form using Eq. (2.16) as 1 * 1 QQ b b S K E ( K) S ( K) E ( K), (2.36) where E * is the complex conjugate transpose of matrix E. The PSD of the physical displacements (Eqs. 2.1a 2.1c) can be obtained from the PSD of the respective generalized displacement components through S x, x, K B h x h x S K, (2.37) 2 hh A B i A j B i j i j

42 26 S x, x, K B p x p x S K, (2.38) 2 pp A B i A j B i j i j S x, x, K x x S K, (2.39) A B i A j B i j i j where i and j are summed as the summation over the number of modes being used. Crossspectral densities can be developed in a similar manner. Evaluation of the spectral densities of the displacements at combinations of discrete xa and xb will result in a matrix. The mean-square values of these displacements can be evaluated in terms of their respective PSD functions. 2 2 B hh S, 0 hh K dk U (2.40a) 2 2 B pp S, 0 pp K dk U (2.40b) 2 2 B S K dk, U 0 (2.40c) where K is the reduced frequency. A covariance matrix for h, p and α is thus obtained, from which statistics of the displacement components h, p and α can be calculated. A use of the reduced frequency, K, as the variable of integration results in an additional factor of 2πB/U in the estimation of mean-square value.

43 Effect of Wind Directionality: Skew Wind Theory In the previous sections, considering the wind as it approaches the bridge orthogonal to the deck axis derives basic flutter and buffeting studies. However, in nature, the highest winds of record at a given site is very likely to be skew to the bridge (Scanlan 1999). Therefore, flutter and buffeting analyses are modified to account for directionality θ (mean-wind yaw angle in Fig. 2.3), as originally proposed in (Scanlan 1993). A review of skew wind buffeting theory is presented in this section. The material is derived from (Scanlan 1993). In Fig. 2.3, the plan view of the deck is shown with steady mean wind velocity U and accompanying turbulence components (i.e., u(t), v(t) and w(t)) and approaching the bridge at an angle θ with respect to the direction orthogonal to the deck axis indicated by the normal y (or y(n) in Fig. 2.3). As explained in (Scanlan 1993), the effect of a skew wind on the bridge vibration can be estimated by considering a component Un normal (across-deck) to the deck of mean wind velocity and two turbulence components u and v as U U ucos v sin, (2.41) n U U usin v cos. (2.42) p The effect of the parallel (along-deck) component to the deck Up, usually of minimal relevance, was neglected in this research. As an example, vertical and torsional aeroelastic loadings due to two components of skew wind U shown in Fig. 2.3 can be constructed by replacing the original expressions by means of a reduced skew frequency Kn:

44 28 L ae 1 2 U 2 * h B K n H 1 U K H * B n 2 U K H n 3* K 2 * h n H 4 B (2.43) M ae 1 2 U 2 B 2 * h K n A 1 U K A * B n 2 U K A n 3* K 2 * h n A 4 B (2.44) where given the reduced K = Bω/U, the normal component becomes Kn = K/cosθ. Flutter derivatives in previous equations are also evaluated at Kn (sometimes referred to as the cosine rule ). Similarly, buffeting loads can be constructed due to skew wind effect as follows: 1 2 u dcl w vt Lb U Bcos2CL CD 2CLsin, 2 U d U U u dcm w vt Mb U B cos2cm 2CMsin, 2 U d U U (2.45) (2.46) with static coefficients evaluated at α0, CD, CL and CM. This formulation can be recast in Eqs. ( ) as a function of u, v, w, θ and Kn and can be used directly into the framework of the multimode algorithm, introduced in the previous sections.

45 29 Figure 2.1 A suspension bridge and a section of the deck (Schematic view of a generic finite-element model of the structure). Figure 2.2 Degrees of freedom and aeroelastic forces on a bridge deck (the p component neglected in this study).

46 30 y (n) Bridge Deck x B U p U n θ U+u(t) w(t) v(t) Figure 2.3 Schematic plan view of bridge deck with skew wind approaching the girder at wind speed U with turbulence components u, v, w and skew wind angle θ (Scanlan 1993).

47 Chapter 3 A Second-order Polynomial Model for Flutter Derivatives This chapter introduces the concept of model curve, a polynomial-based function for flutter derivatives in terms of reduced velocity and used to describe in a generalized form; the coefficients of this polynomial are random variables considering the uncertainty in the flutter derivative (FD). Probability distribution or the random variables is conditional on the reduced wind speed. For computational reasons in subsequent analysis, however, this dependency is neglected and the probability of these random variables is treated as independent of the reduced wind speed. For analysis purposes the first- and second-order statistics are estimated from experiments, treating all the wind speed data as part of the same population. Experiments were conducted in the wind tunnel, maintained by the Department of Mechanical and Industrial Engineering at Northeastern University (NEU). This chapter describes the experimental methods and wind tunnel tests, employed for the extraction of aeroelastic coefficient or flutter derivatives (FDs) and also for the 31

48 32 estimation of the first- and second-order statistics of the polynomial model. Section models, representing a section of the deck in a long-span bridge, were used in the study. These models are intended to represent only a portion of the deck of the bridge, i.e., the section schematically shown in Fig The description of the wind tunnel and bridge models used in this study are given. The FDs were found simultaneously from two-degree-of-freedom (two-dof) coupled motion section model tests. The model curve will be later used in Chapters 4 and 5, which enables to directly project the uncertainty in the FD into the analysis of the buffeting response of the bridge 3.1 Description of the Polynomial Model and Discussion on its Physical Interpretation Description of the Polynomial Model It has been shown (e.g., Scanlan and Tomko, 1971) that most flutter-derivative experimental curves tend to follow a similar trend, especially for relatively bluff deck sections (Simiu and Scanlan 1996). Postulating a second order polynomial model for flutter derivatives (FDs) to describe the evolution of flutter derivatives as a function of reduced velocity was proposed as a physically acceptable assumption in the context of simulation. The polynomial, labeled as model curves, for FDs as a function of reduced velocity UR=2π/K with i = 1,...,4 are shown below: H U C U C U i 1,..., 4 j 1,3,5,7 (3.1) * 2 i R j R j 1 R, A U D U D U i 1,..., 4 j 1,3,5,7 (3.2) * 2 i R j R j 1 R.

49 33 These parameters become Cj and Cj+1 for Hi *, Dj and Dj+1 for Ai *. The general form for all Hi * and Ai * derivatives can be expressed as in Eqs. ( ) with i=1,,4 and j=1,3,5,7. In Eq. (3.1), Cj and Cj+1 are constant parameters of the model, which are assumed as random coefficients and can be related in a simple way to experimental errors. The mean values of Cj and Dj can be determined from the mean of experimental points, extracted at various wind speeds in wind tunnel; similarly, second-moment properties of Cj and Dj can be related to the variances of measured FDs; the coefficients of the polynomial are determined from statistical regression of the experimental data, described in a separate subsection Discussion on the Selection of the Polynomial Model, based on Physical Behavior of Flutter Derivatives It must be noted that the selection of the model curves, based on a second order polynomial, is not arbitrary but has a direct interpretation with the physical phenomenon related to the concept of FD. FDs are employed to describe the unsteady fluid-structure interaction. Nevertheless, as a first approximation, the Hi * and Ai * coefficients can be estimated by using a suitable combination of the static lift and moment coefficients of the deck section model, measured in a static test of the model, rigidly mounted on a fixed force balance (e.g., Simiu and Scanlan, 1996). Using the general theory of quasi-stationary wind forces (e.g., Simiu and Scanlan, 1996) and recalling that air inertial contributions are negligible in this formulation (whence H * 4, A * 4 derivatives cannot be evaluated), the approximate expressions

50 34 of the flutter derivatives as a function of reduced frequency K=2π/UR for initial angle of attack α0 close to zero are (Singh 1997; Strømmen 2006) dc L dc L dc L d * 0 0 d * 0 0 d * 00 H1, H2, H3, 2 K K K (3.3a) dc M dc D dc M d * 0 0 d * 0 0 d * 00 A1, A2, A3. (3.3b) 2 K K K In the expressions above the derivation with respect to the static angle of attack α is applied to the static lift coefficient (CL) and moment coefficient (CM), which are constant and independent of flow speed. Expressions above are inversely proportional to K (with exponent at most equal to 2) or, in other words, proportional to the reduced velocity UR with the same exponent. The equations above show that flutter derivatives can be theoretically interpreted as a monomial in terms of reduced velocity (inverse of K), which is at most of order two. Since the expressions above are theoretically valid for low K only (K of the order 0.2 to 0.4) only (Strømmen 2006) it is reasonable to assume a polynomial expression as the most plausible model for the derivatives, based on physical evidence. Therefore, the use of Eqs. (3.1) and (3.2) in the model curves is justified by physical evidence. This interpretation is, however, valid for linear superposition of aeroelastic effects, because of the use of the static coefficients in Eq. (3.3) and for small vibrations h or α compared to the reference dimension of the deck (width B or depth D).

51 Description of the Wind Tunnel, used for Experimental Verification of the Polynomial Model The wind tunnel tests were performed in the small-scale low-speed wind tunnel of Northeastern University (NEU). A two-dof elastic force balance had been designed by a former graduate student (Brito 2008) and built for free-vibration tests of scaled models of bridge deck sections. The design of the balance was partially based on an existing setup, developed by (Chowdhury and Sarkar 2003). The design of the NEU setup was tailored to the specific characteristics of the NEU s wind tunnel due to the limitation of the physical dimensions of the facility and of the test chamber. The experimentation was based on a section-model, which is a replica of a section of the actual bridge. Section model consists of representative span-wise sections of the deck constructed to scale, spring supported at the ends to allow for both vertical and torsional motion (Simiu and Scanlan 1996). These are constructed at a reduced geometric scale in comparison with the full-scale structure, usually of the order 1:50. Section models are widely used since they have the important advantage of enabling the measurement of the fundamental aerodynamic and aeroelastic characteristics of the bridge deck (flutter derivatives). The wind tunnel tests were carried out in a closed circuit wind tunnel. The tunnel has a 305 mm 305 mm (12 in 12 in) test section and produces wind speeds up to 45 m/s (150 ft/sec) in smooth flow. The air flow is driven by a 15 hp DC motor which is connected to a compressor blade that generates the air flow. The generated flow is controlled to pass from

52 36 the settling chamber toward the drive section, where the air is recalculated (see Fig. 3.1, (Brito 2008)). This motor provides sufficient power to move the air though the tunnel. The wind tunnel was originally designed for mechanical engineering applications (i.e., for testing in smooth flow) so that it does not provide a long test section. For civil engineering application the long test section is desirable to simulate atmospheric turbulence (i.e., for simulating the boundary layer flow). However, it was not necessary, since smooth flow was used in the tests. In fact, the purpose of this study was to interpret uncertainties on measurement errors. Turbulent flows and boundary layer flows for section model analysis of bridges can be experimentally obtained by simply adding passive devices (e.g., girds and honeycomb mesh) at the exit of contraction cone, which can generate a uniform turbulence field in front of the deck section. Uniform turbulence is acceptable in the contest of section model testing of bridge decks (Jones and Scanlan 2001). 3.3 Description of the Experimental setups, used for Verification The experimental setup allows for two-dof free vibration test simulating vertical (h) and torsional (α) dynamic response simultaneously at different wind velocities. Lateral displacements (p) were not considered as these usually affect marginally the dynamic response. This section briefly describes the setup, designated as NEU s Aeroelastic Force Balance (i.e., shown in Fig. 3.2), developed by former graduate student (Brito 2008). The setup includes a T-shaped rig, mounted externally to the wind tunnel test chamber ( the box ), for the suspension of the model, which can be vertically supported on a set of extension springs. The springs were selected such that the spring combination,

53 37 configuration and spacing, for a particular DOF (either h or α) could produce the desired stiffness (i.e., frequency) in the two-dimensional dynamic system calibrated to obtain a reasonable duration of time history response in the free-decay tests. The target mechanical frequencies of the dynamic system, selected for the design, were chosen as about 6 Hz and 10 Hz for vertical and torsional DOF, respectively. The extension springs can be mounted at pre-selected distances from the center of the section model, allowing changes in the frequency ratio between torsional and vertical oscillation of the section model. Initial pretensioning in the springs was imposed to ensure that large displacements, even two to three times the depth of the model, were possible during behavior in the mechanical model (e.g., slackening of the springs). 3.4 Description of the Aeroelastic Section-Model, used for Verification The girder of the Golden Gate Bridge (truss-type deck girder shown in Fig. 3.3) was selected as the benchmark in this work. A bridge section model, based on information derived from (Jain 1996; Jain et al. 1996; 1998) was built at a geometric scale 1:360 and used in the wind tunnel investigations; data from the experiments are later employed in the simulations in Chapters 4 and 5. The Golden Gate Bridge (full-scale structure) is a suspension bridge over the Golden Gate, a strait between San Francisco Bay, California (USA) and the Pacific Ocean. It connects the city with Marin County. The bridge consists of a center span of 1,280 m (4,200 ft) and deck width of 27.5 m (90 ft). The cables are each supported on two steel towers,

54 38 each rising 227 m (746 ft) above water level, and are anchored in massive concrete anchorage blocks at their ends. Figure 3.2(b) depicts the section model of the deck used in this study, mounted on the NEU s balance. This model simulates the aeroelastic behavior of a truss-type deck girder and approximately replicates the features of the Golden Gate Bridge at the scale 1:360. The width-to-depth aspect ratio for the deck girder is B/D = 3.5:1; the geometric scale selection was dictated by the maximum dimensions of the test chamber at NEU; the model scale was kept as a minimum to avoid blockage effects, which would have required corrections of the aeroelastic forces and flutter derivatives (Barlow et al. 1999). The mechanical frequencies of the experimental apparatus (which include the mass of the model and of the rig) are: 7.8 Hz (vertical) and 9.8 Hz (torsional); model width is B = 76 mm. 3.5 Description of the Tests and Experimental Identification Eight flutter derivatives of this truss-type bridge section model (Fig. 3.5), H1 *,,H4 * (for lift force) and A1 *,,A4 * (for moment), were extracted from 1-DOF and 2-DOF free-decay dynamic tests; the setup is shown in Fig. 3.2(b). The Iterative Least Squares Method (ILS) (Chowdhury and Sarkar 2003) was used for identifying the flutter derivatives for lift and moment. The ILS method was developed for the extraction of flutter derivatives from free vibration displacement time histories obtained from a section model testing in the wind tunnel. The main features of the method are briefly summarized below. The equations of motion for the section model, subjected to aeroelastic forces, can be written as a function of the state vector h, T y as

55 39 My Cy Ky F ae. (3.4) The mass matrix M, damping matrix C and stiffness matrix K of the mechanically suspended system (Fig. 3.2b) are of dimension 2 by 2 and can be assumed as diagonal if the mechanical coupling between the degrees-of-freedom h and α is eliminated by appropriate design of the suspension system and model in the experimental setup. The scalar terms of the mass matrix include the contribution of the moving masses and suspension system. These terms can be determined from a series of preliminary experiments in the absence of wind flow as described in Brito (2008). The vector of aeroelastic forces Fae in Eq. (3.4) includes lift force and moment, which are related to flutter derivatives in Eqs. (2.6a) and (2.6c), respectively. Since Fae is a linear function of h and α and their time derivatives (i.e., a linear function of y and y), Eq. (3.4) can be rewritten in terms of aeroelastically modified or effective damping and stiffness matrices, as in Eq. (3.5b), by eliminating dependence on the force vector and after mass rescaling i.e., pre-multiplication by the inverse of the mass matrix (Eq. 3.5a). If this interpretation of the fluid-structure interaction phenomena is employed, the effective damping and stiffness matrices include both the contribution of mechanical part and aeroelastic one. The aeroelastic part is isolated from the mechanical part by repeating the measurements in the absence and in the presence of the flow at a pre-selected speed U. The equations of motion for the section model subjected to aeroelastic forces can be written as

56 40 y M 1 Cy M 1 Ky M 1 F ae, y C Mech y K Mech y M 1 F ae. (3.5a) Or, after rearranging the terms y C eff y K eff y 0. (3.5b) Equation (3.5b) can be expressed in the state space as y y 0 I K eff C eff A y y. (3.6) The A matrix contains the aeroelastically modified effective damping and stiffness matrices, C eff and K eff, respectively. I is the identity matrix of size 2 2. The A matrix can be identified if displacement data are recorded and accelerations and velocities estimated by numerical differentiation; the records should include all n=2 degrees of freedom for at least 2n different instants of time (Ibrahim and Mikulcik 1977). In accordance with the ILS Method, tests are conducted at a given flow speed U to extract the aeroelastically modified effective damping and stiffness matrices, C eff and K eff. Tests are also repeated at U=0 for replicating free vibration without air flow to extract the mechanical matrices K mech, C mech. The flutter derivatives can be determined from the

57 41 difference (K eff K mech ) and (C eff C mech ), as indicated in (Chowdury and Sarkar 2003, Brito 2008). For example, H1 * and A3 * are given as (Chowdury and Sarkar, 2003) * 2m h eff mech 2 H K C C, B (3.7) * 2I eff mech A3 K 4 2 K22 K22. B (3.8) 3.6 Reason for the Use of the Polynomial Model in the Context of Random Flutter Derivatives In the presence of measurement errors, flutter derivatives Hi * (K) and Ai * (K) become random functions in terms of K = (2π)/UR (or, equivalently, reduced velocity). The results of the experiments (later discussed) suggested that the variance of the derivatives, experimentally estimated by repeating measurements at the same reduced velocity in the wind tunnel (i.e., through sample averaging ), can be a function of reduced wind speed. Therefore, it is plausible to also conclude that the probability distribution of each Hi * (K) and Ai * (K) may be different at the various K (or reduced velocities). Since the evaluation of PSD matrix of the generalized buffeting response is based on Eq. (2.36), i.e., 1 * 1 ( ) ( ) QQ ( ) b b S K E K S K E K, a random set of flutter derivatives would correspond to a stochastic matrix E (matrix of the aerodynamically-modified frequency response functions ), the statistical properties of which vary with K as a result of the random Hi * (K) and Ai * (K) being dependent on K. The coefficients Hi * (K) and Ai * (K) can in fact be seen as non-uniform random variables in terms of K. The random properties

58 42 of E lead to a stochastic PSD matrix of the generalized response S ξξ, non-uniform in terms of K, which would be computationally very challenging to replicate in a Monte-Carlo setting since the random properties of each Hi * (K) and Ai * (K) (mean, variance and approximate probability distribution) would be needed K by K. Needless to say, even a non-numerical approach, for example by seeking an analytic solution of the stochastic problem by expansion method about a mean solution case, would be very difficult to pursue since the statistical properties would be needed to be specified (and estimated experimentally) at all K values with acceptable fidelity. For these reasons, the use of the polynomial model was preferred since it reduces the complexity of the stochastic problem by condensing the evaluation of the uncertainty in the flutter derivatives (dependent on K) into the randomness of the coefficients of the polynomial model, which are treated independently of K in the proposed model (Cj and Cj+1, Dj and Dj+1 in Eqs. 3.1 and 3.2). The coefficients of this polynomial therefore random variables, whose probability distribution is not conditional on the reduced wind speed by neglecting the dependency on K (or reduced velocity). The advantage is that, for analysis purposes, the first- and second-order statistics of the coefficients can be estimated from the combination of all experiments, treating all the wind speed data in the wind tunnel (or equivalently, the K values) as part of the same population. This fact corresponds to an increment in the size of the population, used to estimate the first- and second-order statistics from experiments, which also leads to a better confidence on the estimates of the statistics from the experiments.

59 Estimation of Variance and Co-variance of Cj and Dj coefficients of the Model Cures from Experiments Second order statistical moments of flutter derivatives (FDs) can be estimated from a sample population, obtained by repeating the measurements at various wind speed U and the identification process in Section 3.5. The variances and co-variances of each random variable pair Cj and Cj+1, Dj and Dj+1 of the model curves in Eqs. (3.1) and (3.2) were indirectly calculated from the second order statistical moments of the FDs, using the experimental data as in Fig As an example, the following equation is valid for the k-th experimental point of flutter derivative H1,k * at a reduced velocity UR,k: E H E C U E C U 2 U E C C. * , k 1 R, k 2 R, k R, k 1 2 (3.9) In Eq. (3.9) the index k = 1,,N is related to a measurement at a given UR,k. Since measurements are repeated at the same (or very close) reduced velocity twenty times, it is possible that UR,K UR,K. The total second moments of C1 and C2 were estimated by regression of Eq. (3.9), as explained below: *2 E H 1, U R,1 U R,1 2U 2 EC R,1 1 2 EC *2 E H 1,N (3.10) U R,N U R,N 2U R,N E C 1 C 2 b R z

60 44 The size of the b vector is N, the total number of measurements. The unknown vector of the moments z (3 1) was calculated from Eq. (3.10) as z = R b, with R being the Moore- Penrose pseudo-inverse of R (i.e., by least squares). Once the quantities in z are determined from Eq. (3.10), estimation of the variances and co-variances of C1 and C2 (mean-removed) is subsequently enabled from the flutter derivative data of H1 *. Similarly, other statistical moments for Hi * and Ai * were determined; specific equations are similar to Eqs. (3.7) and (3.8) but are omitted for brevity. The interval of reduced velocities 5 U R 20, covered by experiments, was considered as acceptable. A probabilistic model for Hi * and Ai * (i=1,,4) (Fig. 3.5) was therefore obtained, as described in Eq. (3.1) for Hi *. The procedure for extracting the total statistical moments is At a given UR (or UR,k), the eight flutter derivatives (H1,k *,,H4,k * and A1,k *,,A4,k * ) were measured in the wind tunnel by repeating experiments and identification procedure multiple times (Brito and Caracoglia 2009); after collection of the data and results, flutter derivatives were treated as independent random variables; The propagation of uncertainty was simulated by treating each of the coefficients in the polynomial expansion in Eq. (3.1) and Eq. (3.2) as random parameters; A total of the sixteen random coefficients, Cj and Cj+1, Dj and Dj+1, were considered; Mutual dependency between the model curve parameters of each flutter derivative (e.g., C1 and C2 of H1 * ) was considered, as described in Eq. (3.9);

61 45 Expectations and second-order moments of the sixteen random coefficients were derived from NEU experiments. It was discovered that a jointly gamma distribution of each pair of dependent random variables Cj, Cj+1 and Dj, Dj+1 was a suitable model to describe the uncertainty found in the experiments of the truss-type deck section model. This assumption was used in the reminder of this study. It must be noted that this selection may be data-driven (e.g., two-type model) and influenced by the type of measurement errors. Other investigations (Bartoli et al. 2009) found that a Gaussian model was acceptable to describe uncertainty in the FD. Nevertheless, the gamma type model were preferable since the probability distributions are sign independent (one-sided) and are therefore more suitable to describe the variability in the Cj, Cj+1 and Dj, Dj+1 due to uncertainty; a change of sign would result in a drastic change of curvature or discrepancy of sign in the polynomial model curve, which would be physically inconsistent with the actual definition of the FDs. This remark can also be clarified by recalling the interpretation of a flutter derivative as an added stiffness or damping effect, induced by fluid-structure interaction. Experimental errors induce a modification to this effect by maintaining a general trend in the behavior of the curves (e.g., the negative curvature in H1 * related to a positive damping effect as UR increases, etc.), whereas a sign reversal or curvature change would be incompatible with the phenomenon.

62 Summary of Experimental Results and Comparison with Literature Data ( Jain s Data ) The first and second order statistics of the coefficients of the model curve were estimated by repeating the tests twenty times (Brito and Caracoglia 2009). Experimental data for H1 *,,H4 *, A1 *,,A4 * vs. reduced velocity UR = U/(nB) are shown in Fig. 3.5 (solid lines with markers); H1 * and H3 * are related to the aeroelastic lift associated with changes in velocity of the vertical DOF (h) and the angular displacement (), respectively; the rest of flutter derivatives correspond to the aeroelastic torque and depend on changes of h and angular DOF velocity ( ) (Eq. 2.6). Flutter derivatives provided from literature were also used in this study, which will be utilized in the estimation of buffeting analysis in comparison with the results with using FDs obtained from NEU. Flutter derivatives, Hi * and Ai * along with the static coefficients of lift and moment at α0 = 0 were derived from (Jain et al. 1998); these are assumed as more reliable data and shown in Fig. 3.6 as a function of reduced wind velocity UR = U/(nB) with n = ω/2π. Static force coefficients CD, CL and CM and their derivatives are also required for buffeting analysis (Chapter 2); the decision was to use the static coefficients measured by Jain (1998) at α0 with both sets of flutter derivative data. The static coefficients and their derivatives at α0 are shown in Table 3.1. The results of the NEU s experiment are very promising since the use of the model curves and the repetition of the tests enabled the characterization of experimental variability and the analysis of the second-order statistics of flutter derivatives (Fig. 3.5 for the examined

63 47 section-model of a truss-type deck). The figure also suggests a nonlinear dependence of the error variance on reduced velocity (wind speed), which was not noted by other investigators. Also, the experimental procedure offers an example of systematic examination and quantification of the variances for most flutter derivatives in a simple way, based on Eqs. (3.9) and (3.10). In spite of the results in Fig. 3.5, the estimation and quantification of such errors is still an open question. In fact very few examples of error estimation are available in the literature; these examples acknowledge this relevance even though they are very limited and have often considered unrealistically simple deck shapes (Sarkar et al., 2009).

64 48 Table 3.1 The static coefficients and their derivatives at α0 (Jain et al., 1998). Static Coefficients C D C L C M dc D /d α dc L /d α dc M /d α

65 Figure 3.1 NEU-MIE wind tunnel (Brito 2008). 49

66 50 (a) Figure 3.2 Experimental setup: (a) NEU s small-scale wind tunnel; (b) NEU s Aeroelastic Force Balance with the truss-type bridge deck model. (b)

67 Figure 3.3 The Golden Gate Bridge (Photo courtesy of Google Image). 51

68 52 D Figure 3.4 Truss-type deck section model, replicated the features of the Golden Gate Bridge at a scale 1:360; model width is B = 76 mm and the aspect ratio is B/D = 3.5:1.

69 53 H 1 * Experimental data -12 Polynomial model U/(nB) (a) H 2 * Experimental data Polynomial model U/(nB) (b)

70 54 H 3 * Experimental data -6 Polynomial model U/(nB) (c) H 4 * Experimental data Polynomial model U/(nB) (d)

71 A 1 * 0-2 Experimental data Polynomial model U/(nB) (e) A 2 * -6-8 Experimental data Polynomial model U/(nB) (f)

72 A 3 * -2-4 Experimental data Polynomial model U/(nB) (g) A 4 * 0-2 Experimental data Polynomial model U/(nB) (h) Figure 3.5 Flutter derivatives of a truss-type section model with aspect ratio B/D= 3.5:1 measured at NEU: (a) H1 * ; (b) H2 * ; (c) H3 * ; (d) H4 * ; (e) A1 * ; (f) A2 * ; (g) A3 * ; (h) A4 *.

73 57 Flutter Derivative, Heave G.Gate H1* G.Gate H2* G.Gate H3* G.Gate H4* U/(nB) (a) Flutter Derivative, Torsion G.Gate A1* G.Gate A2* G.Gate A3* G.Gate A4* U/(nB) (b) Figure 3.6 Flutter derivatives of a truss-type section model(the Golden Gate Birdge) derived from (Jain et al., 1998): (a) heave Hi * (i=1,,4); (b) torsion Ai * (i=1,,4).

74 Chapter 4 A Methodology for the Analysis of Long-Span Bridge Buffeting Response, accounting for Variability in Flutter Derivatives 4.1 Introduction This chapter describes the development of a methodology for predicting the buffeting response of a long-span bridge by Monte Carlo (MC) methods (Robert and Casella 2004; Tempo et al. 2005). In the standard buffeting analysis (labeled as the deterministic case in this work) the result is the value of the RMS dynamic response at a given wind speed. In the proposed probabilistic setting (labeled as statistical case in this work) one estimates the probability that a given threshold for the variance of the response is exceeded. A Monte-Carlo-based methodology is proposed and implemented for predicting the buffeting bridge response and for evaluating the variability due to uncertainty in the FDs ( statistical buffeting analysis). To accomplish these tasks, a second order polynomial model ( model curve ) for the FD is utilized. The model curve is a second order polynomial 58

75 59 description of the FDs where uncertainty is associated with coefficient of the polynomial. The coefficients of this polynomial are treated as random variables, whose probability distribution is conditional on reduced wind speed. For computational reasons in subsequent analysis, however, this dependency is neglected and the probability of these random variables is treated as independent of the reduced wind speed. For analysis purposes the first and second order statistics are estimated from experiments, treating all the wind speed data as part of the same population. A numerical procedure for multi-mode buffeting response ( deterministic case ) was initially developed and its accuracy was validated by comparing with more reliable data from the literature. In the standard multi-mode buffeting analysis, the power spectral density (PSD) of the buffeting loads needs to be computed. This step, carried out by numerical integration, is usually the bottleneck of the multi-mode buffeting analysis method in the modal space. MC and Quasi-Monte-Carlo (QMC) methods were used to numerically compute the PSD of the buffeting loads and to derive the root-mean-square (RMS) dynamic response of a longspan bridge. A benchmark structure was utilized for this purpose (1,200 m suspension bridge). A validation study was carried out by examining the performance of MC integration methods using a series of standard buffeting analyses on the 1,200m bridge (Golden Gate Bridge) and comparing the CPU time on a standard computer; also, the validation was employed to determine the optimal number of sample points required by the MC and QMC procedures in comparison with standard techniques for numerical integration. This part was also based on a series of preliminary investigations (Caracoglia and Velazquez 2007).

76 60 Finally, the complete procedure (labeled as MC-based methodology in this dissertation), which includes the probabilistic setting, is presented and implemented to numerically evaluate for the probability of exceeding a set of pre-selected serviceability thresholds, selected according to the RMS response of the deck as a function of mean wind velocity and mean incident angle (skew wind). Such curves or surfaces were again derived by MC sampling for two simulation examples, based on the same suspension bridge structure (introduced above) and both literature data and measurements at NEU. The MATLAB software environment was employed for coding purposes. The detailed flowchart of the MC-based methodology for statistical buffeting analysis, including the uncertainty in the FD, is shown in Fig. 4.1 and is described in the following sub-sections. 4.2 Multi-Mode Buffeting Analysis ( Deterministic Case ) The closed-form solution for multi-mode buffeting analysis by using the second order polynomial model for FDs was developed in this work. One of the major problems to apply the closed-form solution is related to the numerical integration of Eq. (2.24), which may become computationally demanding and impractical in the context of statistical buffeting. In order to overcome such limitations, Monte-Carlo and Quasi-Monte-Carlo methods (Robert and Casella 2004; Tempo et al. 2005) were introduced and employed for numerical integration.

77 Validation for Closed-Form Solution The benchmark structure was derived from the dynamic behavior of a suspension bridge with main span l = 1200 m, modeled after the Golden Gate Bridge in San Francisco, California (USA). This bridge has a deck width B = 28 m, deck torsional inertia I0 = kg m 2 /m (Jain et al. 1996). Modal structural damping ratio was selected as equal to 0.3% for all modes as a deterministic constant by following the recommendations by Jain et al. (1998) for this bridge. Lateral modes and responses were not analyzed, since the dynamic bridge response for this bridge is mainly controlled by heaving and torsional motions (e.g., Jain et al, 1996). Flutter derivatives, Hi * and Ai * along with the static coefficients of lift and moment at α0 = 0 were reproduced from (Jain et al. 1998); the derivatives are shown in Fig Closed-form (CF) estimation, which employs a standard integration algorithm (i.e., trapezoidal rule) for calculating the double integral in Eq. (2.24) to calculate the RMS response based on standard multi-mode analysis, was employed for examination of the computational efficiency of MC and QMC algorithms. The simulations included up to 10 fundamental structural deck modes of the benchmark structure. The natural frequencies and mode types are shown in Table 4.1. Simplified (sinusoidal-like) mode shapes were assumed and used to describe the main bridge motion (deck and cable vibration) shown in Fig The CF solution was initially validated by comparison with the literature results, provided by (Jain 1996) with 7 modes only. The RMS vibration was calculated for both vertical σhh (Eq. 2.40a) and torsional σαα (Eq. 2.40c) responses at a quarter span of the bridge deck x = l/4 and for wind speed at U = 22.2 m/s shown in Table 4.2. This bridge section was

78 62 selected since it corresponds to the anti-mode of the mode shapes in both the fundamental vertical and torsional modes. The vertical RMS response of the CF solution was in a good agreement with the literature one, since the difference is equal to 4.1% in Table 4.2. Although the difference for an overall torsional RMS displacement was quite large, equal to 182.1% in relative terms, the result was accepted due to the fact that this value corresponds to an actual small rotation in terms of degrees (i.e., angles 0.12 and 0.34 in Table 4.2) at full scale Monte-Carlo and Quasi-Monte-Carlo Methods Monte-Carlo (MC) methods are a class of computational algorithms that rely on repeated random sampling to compute their results. The MC numerical algorithm was employed to compute the PSD of the buffeting loads Eq. (2.24) by integration and, later, to derive the RMS response of the bridge Eq. (2.40) (Smith and Caracoglia 2011). One of the major problems related to the numerical integration of Eq. (2.24), is the fact that the integrand function needed to estimate Shh, Spp, Sαα in Eqs. ( ) must be evaluated numerically over a large portion of (xa, xb) for low frequency K. For low K the whole (xa, xb) space contributes to the S ( K ) function, whereas for moderate or large QQ i j frequencies the non-zero part of the integrand function concerns a tight zone, located along the main diagonal in the (xa, xb) plane. This is a consequence of the decrease of coherence with distance and frequency in the standard multi-mode formulation. Therefore, the assessment of the integral for low-frequency structures becomes numerically demanding since the resonant part of the loading is usually concentrated at low K.

79 63 In the MC-based approach for assessing the double integral in Eq. (2.24), two samples of uniformly-distributed independent random variables are generated within an interval of x, based on actual integration limits in Eq. (2.24) (0 x l) shown in Fig. 4.2(a). These two sets of independent randomly-generated points (Robert and Casella 2004) are used to find an approximation to the generic surface integral in Eq. (2.24). The use of Quasi- Monte-Carlo (QMC) method was also investigated, as a more deterministic version of MC, which is often preferred for reducing the variance of the estimated quantities and improving the numerical efficiency (Tempo et al. 2005). In the QMC method the two-dimensional coordinates of the points were chosen according to a deterministic criterion instead of a random selection to distribute the integration points more evenly. The Halton sequence (Tempo et al. 2005) was used in the QMC algorithm shown in Fig. 4.2(b). An alternative way to reduce computational integration time would be possible if a decomposition of the integral in Eq. (2.24) into resonant and background responses (Davenport 1967) was used. This technique has been originally proposed in (Scanlan and Jones 1990a) for long-span bridge aeroelastic analysis, providing approximate but still acceptable results. Recently, this approach was re-proposed and used in a more generalized form for bridge simulations (Denoël 2009; Gu and Zhou 2009), in an attempt to avoid numerical integration issues. Nevertheless, the full integration is always more precise; it is also preferable for the statistical buffeting analysis because accurate estimation of the probability distribution of the error-contaminated RMS is crucial. Therefore, numerical integration using MC and QMC methods has been chosen in this study.

80 Examination of the Computational Efficiency of the MC and QMC Methods for Calculating the Double Integral in Eq. (2.24) The accuracy and computational efficiency of the MC and QMC algorithms for doubleintegration was assessed by repeating the estimation of the integral in Eq. (2.24) in the absence of uncertainty in FDs ( deterministic case ). Simulations were repeated for various MC integration points (NMC) to select an optimal value of NMC without compromising the algorithmic accuracy. Two-mode analysis for the bridge example, simulating the structural characteristics of existing bridge, was considered in this second investigation, restricted to the first vertical and torsional skew-symmetric modes with frequencies nv1 = Hz and nt1 = Hz (Mode n.2 in Fig. 4.3b and Mode n.8 in Fig. 4.3h). Scatter plots, corresponding to 100 MC-based repeated estimates of the vertical and torsional RMS responses as a function of NMC, were numerically evaluated. The performance of the MC algorithm was also compared against the average relative run time (RT), normalized to the value obtained for NMC = 50,000. The Coefficient of Variation (CoV) was calculated and later employed as an indicator in the selection of the optimal NMC. The maximum relative and bias errors (against the CF target value) were also calculated using both MC and QMC methods. Figure 4.3 shows the scatter plots corresponding to 100 repeated MC simulations with the average RT also being indicated in the plots. The vertical (σhh in Fig. 4.4a) and torsional (σαα in Fig. 4.4b) RMS responses of the deck of the simulated bridge for U = 22.2 m/s were compared to the target value at x = l/4. In Fig. 4.5 the CoV, estimated for each sample population in Fig. 4.4 and corresponding to each value of NMC, is shown in log-log scale. The CoV is a normalized

81 65 measure of dispersion in a distributed sample and was assessed as the ratio between the standard deviation, σy,target, and mean of the sample, Yσ,mean. As shown in Fig. 4.5(a), the CoV was very small for σhh, less than 1.4% in all cases. The CoV for σαα was generally higher but very small for NMC > 10,000, less than 2.0%. In Fig. 4.5(a) and Fig. 4.5(b), the decrement in the CoV was proportional to NMC on the log-log chart with a constant negative slope in the curves; this behavior corresponds to a power-law decrement for increasing NMC on a linear scale. In Table 4.3 the maximum relative and bias errors for MC, estimated for each sample population in Fig. 4.4 and corresponding to each value of NMC, are shown. The bias error was estimated as the difference between the CF value of the RMS response and the mean of the sample, σy,target - Yσ,mean. The relative error is the bias error divided by the mean of the sample, σy,target / Yσ,mean. The relative errors were less than 0.7% for σhh at l/4 and -1.9% for σαα, shown in Table 4.4, while they were almost negligible for larger sample sizes. Tolerance intervals were employed to analyze the fidelity of the numerical algorithms, as shown in Figs. 4.6 and 4.7. These tolerance limits were approximately assessed assuming that the sample of RMS values follows a Gaussian distribution. In the cause of a variable with Gaussian distribution and known population mean, μ, and standard deviation, σ, the following tolerance limits can be used; μ ± z * σ. If z * = 1.96, these bounds (covering the middle 95% of the population of observations) are essentially the confidence intervals for a fixed proportion of the measurements (Walpole et al. 2002). When μ and σ are unknown, which is typically the case, Eq. (4.1) can be applied. Y.,mean k y,target (4.1)

82 66 The quantity k is the tolerance factor for a normal distribution. In this study, k is defined such that there is a 99% confidence that the calculated tolerance limits will contain at least 95% of the measurements, k = The limiting confidence interval (e.g., 99%) must be added to the statement since the bounds given by Eq. (4.1) cannot be expected to contain any specified proportion (e.g., 95%) all of the time (Walpole et al. 2002). The assumption of a normally-distributed population of σh and σα was used. Since the objective was the section of an optimal sample size this hypothesis was accepted. The tolerance interval results of are shown in Figs. 4.6 and 4.7. In both figures, the horizontal axis represents the number of simulations conducted (i.e., 100 simulations) for a given NMC, and the vertical axis is the RMS response. In each plot the tolerance intervals, the target value (from CF simulations) and the results for each of the 100 simulations are presented. All figures show a correlation between the number of integration points, NMC, and the tolerance intervals, since the tolerance interval decreases as NMC increases. The tolerance interval is approximately 9 times larger for NMC = 1,000 than it is for NMC = 100,000. The benefit of using more integration points is clearly shown by the results of this investigation. However, a smaller tolerance interval should not be the only criterion employed for the selection of the optimal NMC. Figures 4.6 and 4.7 also suggest that, as the tolerance interval decreases, the bias error can increase as the difference between the CF target value and the estimated mean grows. Additionally, in Figs. 4.6(d) and 4.7(e) the tolerance limits no longer contain the CF target value. This aspect must also be taken into consideration since the overall goal of the MC simulations is to obtain results close to the CF exact value.

83 67 Figure 4.8 depicts the vertical (σhh in Fig. 4.8a) and torsional (σαα in Fig. 4.8b) RMS responses of the same bridge example, using by QMC integration. Scatter plots and CoV of σhh at l/4 correspond to 100 repeated simulations. The average RT was normalized to the duration of the MC simulation with NMC = 50,000 for comparison with Fig The CoV values in Fig. 4.9 (log-log scale) are much smaller than the ones observed with the MC algorithm for both vertical and torsional response; however, the direct proportionality on the log-log chart with a constant negative slope for all NQMC was not observed. For example, the CoV for NQMC = 5,000 is less than 0.2%, which is clearly smaller than approximately 0.5%, noticeable in Fig. 4.9(a) with NMC = 100,000. However, RT is equal to 5.05, which is three times larger than the MC case with NMC = 100,000 (RT = 2.26). The maximum relative and bias errors for QMC (Table 4.5) are shown. The relative errors in the QMC case were less than 0.9% for σhh at l/4 and -3% for σαα, shown in Table 4.4. The relative and bias errors in the QMC case were larger than the ones in the MC case, even though the QMC case had smaller CoVs. Tolerance intervals were also investigated and are presented in Figs and From the interpretation of Fig. 4.4 through Fig it was concluded that QMC integration, even with a medium sample size (NQMC > 10,000), was impractical because of large RT. The QMC case with NQMC = 1,000 did not improve the numerical performance of the standard algorithm in terms of CoV, with the RT being approximately eleven times larger than the reference case with NMC = 1,000. In the MC case the relative error of σhh consistently increases with an increment in NMC; for the torsional response, the relative error also varies with NMC. The relationship

84 68 between the bias and the CoV was also utilized as an additional criterion for the selection of the algorithm (MC or QMC). The QMC estimation showed smaller maximum relative errors but higher bias errors (shown in Table 4.4). As a consequence of the above observations, the MC integration algorithm was selected in the subsequent stages of this study. By combining the results of Fig. 4.4 through Fig. 4.7, NMC = 5,000 was recommended as the preferable choice for MC due to both relatively small variance and good numerical efficiency. The computational time is approximately one order of magnitude smaller than that of using a standard integration algorithm (repeated trapezoidal rule). 4.3 Monte-Carlo-based Methodology for Buffeting Analysis Considering Uncertainty in the Flutter Derivative ( Statistical Case ) A Monte-Carlo-based methodology is proposed and implemented for predicting the buffeting bridge response and for evaluating the variability due to uncertainty in the FDs simulated by MC sampling. To accomplish these tasks, the second order polynomial model ( model curve in Eq. 3.1 and 3.2) for the FD is utilized, in which the coefficients of the polynomial are random variables; more details can be found in Section In the MC-based methodology for buffeting analysis, the numerical procedure recalculates, at various wind speeds and skew wind angles, the buffeting loads by MC sampling for each of the 5,000 realizations (e.g., the flowchart presented in Fig. 4.1). The generalized power spectral density (PSD) of the buffeting loads is also calculated by MC sampling, where a double integration (Eq. 2.24) is needed. Even though it is not necessary to repeat the double integration for calculating the PSD of the buffeting loads, it is still

85 69 necessary to identify an efficient numerical procedure (Section 4.2.3) in order to be able to generalize the method for future applications. In fact, extension of the method has been investigated, in which also the effects of errors in the buffeting part of the loading must be accounted for. These errors include, but are not limited to, simplifications in the modeling of the wind turbulence spectrum, errors in the estimation of the span-wise loading parameters Suu and Sww in Eq. (2.24). For more information on the effects of this category of errors, the reader may refer to the recent publications in this area (Caracoglia 2008a; 2008b; 2011). In the context of a future generalization of the method, in which the power spectral density of the buffeting loading can become a random function in terms of K, repetition of the buffeting loading estimation is required. Therefore, an efficient numerical procedure was needed. In the proposed probabilistic setting ( statistical buffeting case ) one estimates the probability that a given threshold for the variance of the response is exceeded. There are two obvious formats to display the information that are useful in different ways. One way is to plot the RMS value of buffeting response at a given confidence level of not being exceeded. For example, Fig shows the RMS value for vertical buffeting response as a function of wind speed at a given confidence level. More directly useful way for our purpose is to plot the probability of exceedance at a given fixed RMS value as a function of wind speed. This probability is designated as threshold exceedance probability (TEP) in this work, derived by using a MC-based methodology with 5,000 sampling points (e.g., the flowchart presented in Fig. 4.1) which projects the variability in the FD into the estimation of buffeting response. The concept of using TEP was adopted from seismic engineering field (i.e., fragility ).

86 70 Fragility analysis is a standardized methodology, utilized for performance-based structural design. As a general statement, fragility curves measure (or quantify) the overall structural vulnerability (Norton et al. 2008). The likelihood of structural damage due to different demand levels mean wind velocity levels in the case of wind engineering is usually expressed by a fragility curve (Saxena et al. 2000). A collection of these curves describes the (conditional) probability of exceedance of representative structural response indicators ( structural capacity ), corresponding to a specific feature of the dynamic response at a given wind velocity (Bashor and Kareem 2007; Ellingwood 2000; Filliben et al. 2002). A set of thresholds is usually selected to represent different levels of structural performance derived from such indicators. As an example, in the case of a building these indicators are either required or are prescribed by the designer, and can include inter-story drift ratios, maximum lateral drift, and acceleration levels for occupant comfort (Bashor and Kareem 2007; Filliben et al. 2002; Smith and Caracoglia 2011). The TEP curves were developed in this dissertation by numerically deriving the histogram of occurrences and the subsequent probability density function (PDF) of the RMS dynamic response by a recursive procedure. The RMS response is a random variable in the probabilistic setting. The probability of exceedance of pre-selected thresholds was later calculated. Equation 4.2 below relates the loss of performance of the structure, a bridge in this study, which is the probability of exceeding of a threshold T, associated with the dynamic response feature Y (performance indicator) at a given deck section for wind incidence angle (θ) orthogonal to the longitudinal axis (initially assuming this response independent of wind

87 71 direction) as an exclusive function of mean wind velocity at deck level U (e.g., (Ellingwood et al. 2004; Filiben et al. 2002)). P F Y T U u f udu. (4.2) T T U 0 In Eq. (4.2), the conditional probability function, denoted as FT[], is the TEP for threshold T; fu is the probability density function (PDF) of the mean wind velocity, which can be derived from site wind data under the conservative assumption of constant wind direction, always orthogonal to the longitudinal deck axis. In this section the mean wind velocity at deck level was assumed as being always perpendicular to the bridge longitudinal axis, as this direction usually corresponds to the most unfavorable condition. The combined influence of mean wind speed (U) and incidence angle (θ) is discussed in Chapter 4.5. These were derived after numerically assessing the probability distribution of the RMS dynamic response at representative wind velocities (U). In summary, the numerical procedure, which utilizes this methodology, combines the estimation of RMS response via MC integration with brute-force uncertainty simulation due to flutter derivative errors to estimate the RMS response and TEP curves. The complete flow chart of the procedure with more details is shown in Fig The RMS vertical and torsional response were utilized as an example, noting that the RMS response can be directly related to the peak displacement through gust effect factor (e.g., (Scanlan and Jones 1990a)) for serviceability analysis due to stationary winds (also refer to Chapter 5 for discussion on peak response).

88 Description of the Bridge Example and RMS Threshold Levels ( Probabilistic Setting ) One bridge example was selected for threshold exceedance probability (TEP) analysis, the same bridge model discussed in Section 4.2. Four-mode buffeting analysis was carried out by considering the first two vertical (v1 and v2) and torsional modes (t1 and t2); it has frequencies nv1 = Hz and nv2 = Hz, nt1 = Hz and nt2 = Hz. Simplified (sinusoidal-like) mode shapes, shown in Fig 4.2(b, d, g and h) were used to describe the main bridge motion (deck and cable vibration) in both models. For example, as shown in Fig 4.2, the mode shapes of the vertical modes v1 and v2 were assumed as purely flexural, with shapes hv1(x) = sin(2πx/l) and hv2(x) = sin(πx/l), in which l = 1,263 m is the central-span length for this bridge. Similarly, the shapes of torsional modes t1 and t2 for the same model were simulated as purely torsional with αt1(x) = sin(2πx/l) and αt2(x) = sin(πx/l). The second order polynomial model ( model curve ) for flutter derivatives, proposed in Section 3.4 to approximately account for effects of measurement errors in the FD, was employed in the statistical buffeting analysis. Since eight flutter derivatives (H1 *,,H4 * and A1 *,,A4 * ) are measured in wind tunnel as a function of the reduced speed, UR = U/(nB) = 2π/K, experimental data are usually available at discrete points on the UR (or K) axis. In the TEP analysis, the coefficients of the model curves for each flutter derivative were randomly perturbed to simulate the uncertainty in the FD; Hi * = CiUR 2 + Ci+1UR in Eq. (3.1) with i=1,3,5,7 and Ai * = DjUR 2 + Dj+1UR in Eq. (3.2) with j=1,3,5,7. The parameters of Hi * and Ai * were assumed as a set of uncorrelated gamma-type random variables. Description on the selection of this specific probability distribution may

89 73 be found in Section 3.4. The hypothesis of uncorrelated random variables is also described in Section 3.4 The dispersion and shape parameters of the marginal probability of each variable were associated with mean and standard deviation estimates of Hi * and Ai *. Three threshold levels were selected to derive the TEP curves. Table 4.5 shows the threshold levels employed in the TEP analysis for σhh and σαα. Thresholds were based on a median value of RMS displacements of the 5,000 buffeting analysis at wind velocity equal to 20 m/s for the deck section at x = l/4, for example with T1 being equal to 50%, T2 equal to 100% and T3 equal to 150% of the corresponding the median value. For example, threshold T2 corresponds to a dynamic displacement equal to 0.015D (i.e., D is being depth of the deck equal to 7.83 m). TEP analysis was carried out for the bridge example using flutter derivatives from literature (Jain s FD data) and flutter derivatives measured at Northeastern University (NEU) TEP Curves using Literature Data TEP curves, associated with the benchmark bridge model (Golden Gate Bridge) and based on flutter derivatives reproduced from (Jain et al. 1998), were derived. The statistical properties of the coefficients of the model curve were synthetically reproduced since no FD error analysis was available. The coefficients of the Hi * and Ai * model curves (Eqs. 3.1 and 3.2) were assumed as a set of uncorrelated gamma-type random variables. The dispersion and shape parameters of the marginal probability of each variable were associated with the mean and standard deviation estimates of Hi * and Ai. For example, in Fig the Jain s flutter derivatives H1 * and A2 * are depicted; these data were reproduced from the experiments described in (Jain 1996; Jain et al. 1996) and

90 74 from Fig The graphs also show the reference model curves based on the discrete data points, assumed to be average values in the analysis. The dotted lines describe the upper and lower limits of the reference curve that first statistics of the coefficients of the reference curve was synthetically derived from such limits. Since no error analysis was carried out by the investigators in their experiments, the variability (uncertainty) was indirectly estimated in an approximate way, described above. The TEP curves were calculated for both vertical and torsional vibrations at the quarter span of the simulated bridge. The thresholds in Fig. 4.14(a) and Fig. 4.14(b) were selected from Table 4.5. The TEP curves were developed by numerically deriving the histogram of occurrences for each indicator at each wind velocity U, shown in Fig. 4.14(a) and Fig. 4.14(b). The deterministic flutter velocity (DFV) without flutter derivative errors, estimated as equal to 19.7 m/s, is also indicated a vertical dashed line. In Fig. 4.14, the probability of exceeding T1, T2 or T3 increases as a result of a larger perturbation (higher RMS response) for higher U. A small decrement in the TEP curves related to T1 for U > 19.7 m/s in Fig. 4.14(a) may also be explained by potential inaccuracy in the MC buffeting procedure in the proximity of flutter speed, as described below. Beyond the deterministic flutter speed, the TEP curves are expressed as dotted lines due to the onset of dynamic instability beyond a given critical velocity Ucr (deterministic flutter velocity - DFV). Since the buffeting procedure numerically calculates the RMS displacements of both vertical and torsional responses by spectral methods, it may fail by predicting a finite RMS response at U beyond the velocity of the onset of flutter; finite

91 75 vibration amplitudes and stationary responses are compatible with a post-critical flutter regime but linear random vibration cannot be used anymore to estimate the response. In the context of the study of multiple realizations by using a sample random population within a MC algorithm, this issue may be circumvented by conducting a preliminary flutter analysis for each simulation before calculating the buffeting response. Another possibility would be to increase the resolution of the discrete points used to construct the TEP curves in the regions close to the deterministic flutter threshold for U < Ucr. In either case, the numerical efficiency of the procedure is very limited also because, in the context of a MC algorithm, a fully automated computer procedure for flutter analysis and non-trivial solutions of Eξ=0 is not available. In this study the Bayes Theorem was used as a simple yet efficient way to accurately capture the buffeting response more realistically, by accounting for the probability of flutter onset for U values close to Ucr. This approach utilizes the original (or prior ) estimates of exceedance probability, calculated by employing the standard MC procedure (i.e., Fig. 4.14) and performs a simple rescaling of the probability. This operation leads to a modified TEP curve or posterior estimate of exceedance probability. The rescaling of the TEP curves was obtained by applying Bayes Theorem, as described below. prior likelihood PY P T U P ~FlutterY T U Y T U ~ Flutter P ~Flutter posterior marginal likelihood (4.3)

92 76 P True 1.0, if U U ~FlutterY T U False 1.0, if U U cr cr (4.4) The symbol ~ is used in the previous equations to designate the non-occurrence event, i.e., for the generic event A, P[NOT(A)] = P[~A] (flutter has not occurred). In Eq. Flutter (4.3), P YT U is the modified TEP curve ( posterior probability ); P YT U is the prior probability, coincident with the TEP curves before rescaling which are shown in Figs. 4.14(a) and 4.14(b); P Flutter is the non-flutter probability ( marginal likelihood in terms of Bayes Theory) and Ucr is the deterministic flutter velocity. It must be noted that the calculation of the TEP PY T U ~Flutter should only include, among all random events in the probability, those events for which flutter has not occurred; therefore, this probability may still be less than one for U beyond the DFV since it is based on a sub-set of the cases, for which flutter has still not occurred. Equation (4.4) describes the non-flutter probability conditional on the exceedance of the given threshold and can be interpreted in terms of Bayes Theory as the likelihood function. Since the use of the Eq. (4.3) is physically acceptable in the interval of U close to deterministic flutter speed (15< U < 25 m/s), the validity of Eq. (4.3) is restricted to this U interval and unacceptable at low U and very large U. This observation is compatible with the statement that the likelihood function (Eq. 4.4) must be very close to unit value, which means that flutter is likely to occur is a necessary condition for applicability of the re-scaling. Therefore, the true and fake logical statements are used in Eq. (4.4) to designate this

93 77 region of validity of Eq. (4.3). The rescaling, based on Eq. (4.3), is acceptable if the statement in Eq. (4.4) is true and the conditional probability of flutter is 1.0. In regions with U U (DFV value) the rescaling is dubious since the assumption cr on the hypothesis of a likelihood function close to the unit value is not satisfied; the practical implication is that TEP greater than one may be observed. Figure 4.15 depicts the procedure for rescaling based on Eq. (4.3). For example, the discrete points on the initial curve ( prior TEP) in Fig. 4.15(a) PY T U ~Flutter, after multiplication by the likelihood function in Eq. (4.4), is divided by the marginal likelihood function P Flutter (i.e., the discrete points along the non-flutter probability curve) in Fig. 4.15(b) to finally become Fig. 4.15(c) as a modified curve ( posterior TEP). Each curve in Fig was numerically evaluate by using the MC-based procedures and are specific for the investigated case. The region of approximate validity of Eqs. ( ) is highlighted by a hatched area; it must be noted that the denominator in Eq. (4.3) P Flutter is strictly greater than zero in the region of validity (above 40% in Fig. 4.15(b)), thereby confirming that PY T U ~Flutter is not possible and that the proposed rescaling is compatible with the assumptions. in Fig The posterior TEP curves for T2, derived from Fig after rescaling, are shown TEP Curves using NEU s Flutter Derivative Data In order to further examine the performance of the MC-based methodology for TEP analysis, flutter derivatives measured in the NEU s wind tunnel were used. Eight flutter derivatives

94 78 (H1 *,,H4 * and A1 *,,A4 * in Fig. 3.5) were measured; the description of the wind tunnel and experiments were presented in Chapter 3. Second-order statistical moments of flutter derivatives were estimated from the wind tunnel measurements. Details may be found in Section 3.6. Structural properties of the model, the Golden Gate Bridge, were combined with NEU s experimental data in Fig. 3.5 to obtain a new set of TEP curves, based on threshold levels in Table 4.5 shown in Fig For comparison purposes the T2 TEP curve, derived from Jain s flutter derivative data, was also reproduced from Figs. 4.16(a) and 4.16(b) (dotted line, after the rescaling based on Eq. (4.3). These curves from Fig are shifted to the right of the graph for vertical response and shifted to the left for torsion response. This suggests that the variability in the FD play a significant role in the TEP analysis of this bridge example and can reduce the dynamic performance, especially for the vertical DOF (Fig. 4.17a), if the FDs from literature were used. 4.4 Effect of Wind Directionality on Statistical Buffeting Response: TEP Surfaces In nature, the highest winds of record at a given site is very likely to be skew to the bridge (Scanlan 1999). In this section, the multi-mode approach for wind direction orthogonal to the deck axis was modified to account for directionality (θ), as originally proposed in (Scanlan 1993). Background theory is described in Chapter 2. The latter approach enabled the analysis of the combined influence of wind directionality and velocity, as these should be more realistically used in serviceability analysis. This observation led to the derivation of

95 79 TEP surfaces (Filliben et al. 2002; Grigoriu 2002) as a function of various pre-selected threshold levels. Figure 4.18 shows the flutter derivative surfaces corresponding to T2 threshold in Table 4.5 that include wind directionality affecting buffeting response of a bridge deck. In Fig. 4.19, structural properties of the bridge were combined with NEU s experimental data in Fig A skew wind angle, varying from θ = 40 to θ = 40, was considered in this analysis. Some influence on directionality can be observed in Fig. 4.18, especially for σαα. Figures 4.18 and 4.19 confirm that the skew wind angle equal to zero degrees is the most conservative assumption for both vertical and torsional buffeting responses. 4.5 Exploratory Performance Analysis on a Full-Scale Structure In this section, the results in Section 4.4 were applied to the serviceability analysis of the actual bridge structure, from which the structural and aeroelastic model was originally derived. The structural performance was derived by extension of Eq. (4.2) to the case of combined dependency on mean wind speed and direction. The exceedance probability and loss of performance (PT) of the full-scale bridge structure was referred to TEP surfaces in Fig and threshold T2. Equation (4.2) was re-written as follows: 180 P F Y T U u, f u, dud. (4.5) T T U 180 0

96 80 In Eq. (4.5), the conditional probability function, denoted as FT[], is the TEP surface and fuθ is the joint probability density function between mean wind velocity (U) and mean wind incidence angle with respect to the axis of the bridge, which must be defined in the interval 180 θ 180 (θ = 0 orthogonal to the bridge, θ = ±90 parallel to the bridge). The wind data, used to estimate fu(u) and fθ(θ) given the assumption fuθ = fu fθ, was extracted from the historical records of a meteorological buoy close to the coast of California in San Francisco (USA), which is part of the NOAA (National Oceanic and Atmospheric Administration) system and National Data Buoy Center (NOAA Station , Latitude: N, Longitude: W, Shown in Fig. 4.20). This particular meteorological station was selected because the prototype full-scale application (Golden Gate Bridge) is located in proximity of the station. Data are available on-line from Wind speeds and directions at this station are measured using an anemometer located 7.3 meters above mean sea level. The histogram of annual maxima ( peak data shown in Fig. 4.21) was derived from the NOAA annual wind speed maxima over a 16-year period ( ) and was employed to evaluate fu(u). The annual wind speed maxima, obtained from the 16-year period, was fitted to an Extreme Value Type-I distribution with two parameters, i.e., scale equal to 30.0 and location equal to 9.6 (Simiu and Scanlan 1996). The distribution of the mean yaw-wind angle was based on a non-parametric model for the statistical distribution and derived from the histogram of the mean wind direction (azimuthal) recorded by the sensor at elevation 7.3 m, postulating little effects of elevation on directionality. The azimuthal direction was later converted to yaw-wind angle by using actual orientation of the bridge axis.

97 81 Table 4.6 summarizes the results of the performance analysis and shows the estimation of the PT for both RMS vertical and torsional responses (dynamic component only) for the bridge, evaluated at deck section x = l/4. These are respectively labeled as case 1 and case 2 in the table. In both cases the total probabilities of exceeding threshold T2 in Table 4.5(a) are of the order of 2% by considering both wind speed and directionality. Overall, the performance of the bridge is good from the point of view of the serviceability standards. This analysis was completed by estimation of a generalized safety index for the prototype application, defined as β = Ф -1 (1 PT) (Haldar and Mahadevan 2000) with Ф being the standard normal cumulative density function. The safety indices are also shown in Table 4.6; these should be interpreted as being related to serviceability buffeting limit states. Reliability is acceptable, with safety indices above 2 in both cases. Since effect of chord-wise aerodynamic admittance was not considered in the model, a further reduction in the buffeting loading may possibly lead to a decrement in PT; therefore, the results of Table 4.6 can be interpreted as a conservative estimation of reliability and β. Reliability against collapse (e.g., flutter) is not contemplated in this chapter, even though it was indirectly included in the assessment of PYY, T 2 Uu by Eq. (4.3). 4.6 Summary The Monte-Carlo-based methodology for predicting the buffeting response of a long-span bridge, including the uncertainty in the flutter derivative, was developed. The model curve was utilized in the MC-based methodology; this curve is a second order polynomial

98 82 description of the FDs where uncertainty is associated with coefficient of the polynomial. Statistical properties of the polynomial in the flutter derivatives were estimated both from the literature and a set of measurements. Numerical procedures were coded and implemented in MATLAB, using the methodology described in this chapter; the numerical program was employed, to derive the dynamic response of two bridge structures at full scale. The numerical results show that the RMS buffeting response of a long-span cable supported bridge can be estimated (with sufficient accuracy to be practically useful) using a second order polynomial description of the flutter derivatives. The uncertainty in these derivatives in the model is captured by specifying the coefficients of the polynomial as a vector of random variables having a specified mean and covariance (the values of these properties used in the numerical analyses were obtained experimentally). It is found that there is a significant computational advantage in using Monte-Carlo methods for calculating a double integral that arises in the estimation of the generalized buffeting loading, needed by the multi-mode buffeting analysis. The computational time is approximately one order of magnitude smaller than that of using a standard integration algorithm (repeated application of trapezoidal rule). The results also suggest that the proposed MC-based methodology for buffeting analysis might be used in conjunction with other dynamic indicators for analyzing the serviceability in a full-scale structure (e.g., for comfort of the bridge users).

99 83 Table 4.1 Natural frequencies and mode types of Golden Gate Bridge (Jain 1996). Mode No. NEU (Hz) Mode type Modal Integrals [1] G hihi G pipi G αiαi LS 2.63E E E VAS 3.20E E E LAS 1.72E E E VS 1.90E E E VAS 3.40E E E VS 3.40E E E TAS 6.67E E E TS 2.49E E E VAS 1.80E E E VS 2.60E E E-15 Note: L is Lateral, V is Vertical, T is Torsional, S is Symmetric, AS is Anti-symmetric. Note [1]: Modal inertia normalized to one. Table 4.2 Comparison of closed-form numerical solution with literature results at l/4. RMS Response Closed-Form From Jain (CF) (1996) Diff. (%) σ hh (m) σ αα (rad) σαα (deg)

100 84 Table 4.3 Bias and relative errors in the MC case: (a) for heave σhh; (b) for torsion σαα. N MC Bias 10-4 (σ y,target Y σ,mean ) Relative Error (%) 1, , , , , Note: y denotes either h or α N MC Bias 10-3 (σ y,target Y σ,mean ) Relative Error (%) 1, , , , , Note: y denotes either h or α (a) (b)

101 85 Table 4.4 Bias and relative errors in the QMC case: (a) for heave σhh; (b) for torsion σαα. N QMC Bias 10-3 (σ y,target Y σ,mean ) Relative Error (%) 1, , , ,000 N.A. N.A. 100,000 N.A. N.A. Note: y denotes either h or α (a) N Q MC Bias 10-4 (σ x,target x σ,mean ) Relative Error (%) 1, , , ,000 N.A. N.A. 100,000 N.A. N.A. Note: y denotes either h or α (b)

102 86 Table 4.5 Threshold values for σhh and σαα, employed in the TEP analysis with flutter derivatives from the literature. Threshold Response Threshold Label Vertical (m) Torsional (rad) T 1 = 0.5 MD 0.008D T 2 = 1.0 MD 0.016D T 3 = 1.5 MD 0.024D MD: median value. D: deck depth Table 4.6 Study cases used for serviceability on full-scale structure. Case Type of Response Wind Velocity and Direction Data (NOAA Station ) 1 RMS of vertical response at l/4 2 RMS of torsional response at l/4 Exceedance Probability for Threshold T 2, P T 1-year continuous data year continuous data Generalized Safety Index, β

103 87 Build deck section Test to obtain Data Model curves: H i* = C j U R2 +C j+1 U R A i* = D j U R2 +D j+1 U R Use Data to formulate a model for the FD that includes variability Select wind speed (U) and skew angle (θ) Spectrum analysis of buffeting loads, S QQ Calculate aeroelastic loads MC sampling of FD simulation Formulate the Equation of Motion in the frequency domain and incorporate the FD information Estimate the PSD of the response S ξξ F.E.M K, M, C (Deterministic) Estimate the RMS response or hh Note: The numerical procedure depicted in the box is repeated for N MC times at various wind speeds and skew wind angles Figure 4.1 Flowchart describing the MC-based methodology for buffeting analysis.

104 x B x A (a) x B x A (b) Figure 4.2 Two-dimensional sample points = 1,000: (a) MC with uniform distribution, (b) QMC with Halton sequence.

105 89 Modal Amplitude Mode n. 1 p h Longitudinal Abscissa (a) Modal Amplitude Mode n. 2 p h Longitudinal Abscissa (b) Modal Amplitude Mode n. 3 p h Longitudinal Abscissa (c)

106 90 Modal Amplitude Mode n. 4 p h Longitudinal Abscissa (d) Modal Amplitude Mode n. 5 p h Longitudinal Abscissa (e) Mode n. 6 Modal Amplitude p h Longitudinal Abscissa (f)

107 91 Modal Amplitude Mode n. 7 p h Longitudinal Abscissa (g) Modal Amplitude Mode n. 8 p h Longitudinal Abscissa (h) Modal Amplitude Mode n. 9 p h Longitudinal Abscissa (i)

108 92 Modal Amplitude Mode n. 10 p h Longitudinal Abscissa (j) Figure 4.3 Ten simplified (sinusoidal-like) mode shapes used in the multi-mode buffeting analysis: (a) LS, Hz; (b) VAS, 0.087Hz; (c) LAS, Hz; (d) VS, Hz; (e) VAS, Hz; (f) VS, Hz; (g) TAS, Hz; (h) TS, Hz; (i) VAS, Hz; (j) VS, Hz. (Note: L is Lateral, V is Vertical, T is Torsional, S is Symmetric, AS is Anti-symmetric).

109 93 RMS Response, σ hh (m) RT: 1.00 RT: 2.26 RT: 0.21 RT: 0.13 NMC= N =1,000 NMC N =5,000 = NMC N =10,000 = RT: 0.05 NMC N =50,000 = NMC N =100,000 = Target Value from CF (RT: 1.46) Note: Relative Run Time (RT) to N MC = 50,000 (a) RMS Response, σ αα (rad) NMC N =1,000 = NMC N =5,000 = NMC N =10,000 = NMC N =50,000 = NMC N =100,000 = Target Value from CF (RT: 1.46) RT: 1.00 RT: 2.26 RT: 0.13 RT: 0.21 RT: 0.05 Note: Relative Run Time (RT) to N MC = 50,000 (b) Figure 4.4 MC-based scatter plots of RMS response for deck section at x= l/4 and for U = 22.2 m/s as a function of NMC: (a) heave σhh; (b) torsion σαα.

110 % CoV, σ hh 1.0% 0.1% 0.0% 1,000 10, ,000 N MC (No. of MC Integration Points) (a) 10.0% CoV, σ αα 1.0% 0.1% 0.0% 1,000 10, ,000 N MC (No. of MC Integration Points) (b) Figure 4.5 Coefficient of variation of the RMS response at x = l/4 for U = 22.2 m/s, computed by MC algorithm: (a) heave σhh; (b) torsion σαα.

111 NMC N = 1,000 Target Value from CF Tolerance Limits RMS Response, σ hh (m) MC Simulation Index (a) NMC N = 5,000 Target Value from CF Tolerance Limits RMS Response, σ hh (m) MC Simulation Index (b)

112 NMC N = 10,000 Target Value from CF Tolerance Limits RMS Response, σ hh (m) MC Simulation Index (c) RMS Response, σ hh (m) NMC N MC = 50,000 Target Value from CF Tolerance Limits MC Simulation Index (d)

113 NMC N = 100,000 Target Value from CF Tolerance Limits RMS Response, σ hh (m) MC Simulation Index (e) Figure 4.6 Tolerance intervals for vertical RMS response (σhh) of 100 MC simulations: (a) NMC = 1,000; (b) NMC = 5,000; (c) NMC = 10,000; (d) NMC = 50,000; (e) NMC = 100,000.

114 NMC N = 1,000 Target Value from CF Tolerance Limits RMS Response, σ αα (rad.) MC Simulation Index (a) NMC N = 5,000 Target Value from CF Tolerance Limits RMS Response, σ αα (rad) MC Simulation Index (b)

115 99 RMS Response, σ αα (rad.) NMC N = 10,000 Target Value from CF Tolerance Limits MC Simulation Index (c) RMS Response, σ αα (rad.) NMC N = 50,000 Target Value from CF Tolerance Limits MC Simulation Index (d)

116 100 RMS Response, σ αα (rad.) NMC N = 100,000 Target Value from CF Tolerance Limits MC Simulation Index (e) Figure 4.7 Tolerance intervals for torsional RMS response (σαα) of 100 MC simulations: (a) NMC = 1,000; (b) NMC = 5,000; (c) NMC = 10,000; (d) NMC = 50,000; (e) NMC = 100,000.

117 RMS Response, σ hh (m) RT: 0.57 RT: 5.05 RT: NQMC N = = 1,000 NQMC N = = 5,000 NQMC N = = 10,000 Target Value from CF (RT: 1.46) Note: Relative Run Time (RT) to N MC = 50,000 (a) RMS Response, σ αα (rad.) NQMC N = = 1,000 NQMC = = 5,000 NQMC = = 10,000 Target Value from NI RT: 5.05 RT: RT: 0.57 Note: Relative run time (RT) to N MC = 50,000 (b) Figure 4.8 QMC-based scatter plots of RMS response for deck section at x= l/4 and for U = 22.2 m/s as a function of NQMC: (a) heave σhh; (b) torsion σαα.

118 % CoV, σ hh 1.0% 0.1% 0.0% 1,000 10, ,000 N QMC (No. of QMC Integration Points) (a) 10.0% CoV, σ αα 1.0% 0.1% 1,000 10, ,000 N QMC (No. of QMC Integration Points) (b) Figure 4.9 Coefficient of variation of the RMS response at x = l/4 for U = 22.2 m/s, computed by QMC algorithm: (a) heave σhh; (b) torsion σαα.

119 NMC N QMC = 1,000 Target Value from CF Tolerance Limits RMS Response, σ hh (m) QMC Simulation Index (a) NNMC QMC = 5,000 Target Value from CF Tolerance Limits RMS Response, σ hh (m) QMC Simulation Index (b)

120 NNMC QMC = 10,000 Target Value from CF Tolerance Limits RMS Response, σ hh (m) QMC Simulation Index (c) Figure 4.10 Tolerance intervals for vertical RMS response of 100 MC simulations (σhh): (a) NQMC = 1,000; (b) NQMC = 5,000; (a) NQMC = 10,000.

121 NMC N QMC = 1,000 Target Value from CF Tolerance Limits RMS Response, σ αα (rad.) QMC Simulation Index (a) NMC N QMC = 5,000 Target Value from CF Tolerance Limits RMS Response, σ αα (rad.) QMC Simulation Index (b)

122 106 RMS Response, σ αα (rad.) NMC N QMC = 10,000 Target Value from CF Tolerance Limits QMC Simulation Index (c) Figure 4.11 Tolerance intervals for RMS response of 100 MC simulations (σαα): (a) NQMC = 1,000; (b) NQMC = 5,000; (a) NQMC = 10,000.

123 107 RMS Vertical Response (m) % Confidence Level 98% Confidence Level U (m/s) Figure 4.12 RMS values for vertical buffeting response as a function of wind speed U corresponding to a given confidence level.

124 108 Flutter Derivative, Lift Flutter Derivative, Moment G.Gate H1* Reference Reduced Velocity (a) G.Gate A2* Reference Reduced Velocity (b) Figure 4.13 Flutter derivatives H1 * (a) and A2 * (b) of the Golden-Gate Bridge girder with aspect ratio B/D = 3.5:1. Data sets were reproduced from (Jain 1996; Jain et al. 1996) with α0=0. The ( reference ) coefficients of the Polynomial Model were derived by regression of the data sets, according to Eqs. (3.1) and (3.2). Tolerance limits (dashed lines) were based on approximate evaluation of one standard deviation.

125 109 1 Probability of Exceedance DFV T 1 T 2 T U (m/s) (a) 1 T 1 Probability of Exceedance T 2 T 3 DFV U (m/s) (b) Figure 4.14 TEP curves of RMS response with respect to thresholds T1 to T3 at the deck section l/4: (a) σhh; (b) σαα (DFV: Deterministic Flutter Velocity).

126 P[(Y>T)] DFV U (m/s) (a) P[~Flutter] DFV U (m/s) (b)

127 111 1 P[(Y>T) ~Flutter] DFV U (m/s) (c) Figure 4.15 Procedure for rescaling the TEP curves in Fig. 6.11(c) based on Eq. (6.3): (a) prior probability or TEP; (b) marginal likelihood function; (c) Posterior probability or TEP (DFV: Deterministic Flutter Velocity).

128 112 1 Probability of Exceedance DFV 0.2 T 2 T 2 M U (m/s) (a) 1 Probability of Exceedance DFV 0.2 T 2 T 2 M U (m/s) (b) Figure 4.16 TEP curves of RMS response at deck section l/4 (T2 threshold only) before (T2) and after rescaling (T2M): (a) σhh; (b) σαα (DFV: Deterministic Flutter Velocity).

129 113 1 Probability of Exceedance T T 3 T 2 M U (m/s) (a) T 1 Probability of Exceedance T 1 T 2 T 3 T 2 M U (m/s) Figure 4.17 TEP curves of RMS responses with thresholds based on the RMS displacement, deck section at l/4 and NEU s flutter derivatives: (a) σhh; (b) σαα. (b)

130 114 Probability of Exceedance (deg) U (m/s) (a) Probability of Exceedance (deg) U (m/s) (b)

131 115 Probability of Exceedance (deg) U (m/s) (c) Probability of Exceedance (deg) U (m/s) (d) Figure 4.18 TEP surfaces of RMS displacement for T2M threshold as a function of wind accounting for effects of skew winds at l/4 with literature flutter derivatives: (a) σhh at l/4; (b) σαα at l/4; (c) σhh at l/2; (d) σαα at l/2.

132 116 Probability of Exceedance (deg) U (m/s) (a) Probability of Exceedance (deg) U (m/s) (b)

133 117 Probability of Exceedance (deg) U (m/s) 40 (c) Probability of Exceedance (deg) U (m/s) 40 (d) Figure 4.19 TEP surfaces of RMS displacement for T2 threshold as a function of wind accounting for effects of skew winds at l/4 with NEU s flutter derivatives: (a) σhh at l/4; (b) σαα at l/4; (c) σhh at l/2; (d) σαα at l/2.

134 118 Figure 4.20 National Data Buoy Center (NOAA Station , Latitude: N, Longitude: W) (Photo reproduced from NOAA,