An experimental validation of a band superposition model of the aerodynamic forces acting on multi-box deck sections

J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

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An experimental validation of a band superposition model of the aerodynamic forces acting on multi-box deck sections G. Diana, D. Rocchi, T. Argentini n Politecnico di Milano, Department of Mechanical Engineering, via La Masa 1, 20156 Milano, Italy

a r t i c l e i n f o

a b s t r a c t

Article history: Received 29 March 2012 Received in revised form 19 December 2012 Accepted 22 December 2012 Available online 23 January 2013

Nonlinear aerodynamic effects on a multi-box deck section are both numerically and experimentally investigated. The basic hypothesis of the band superposition approach, a methodology to model the aeroelastic and buffeting terms of the wind loads acting on a bridge deck, is validated through specific wind tunnel tests on a sectional model. An improved version of the band superposition approach is proposed, introducing a rheological model for high frequency unsteady forces. The numerical model is validated against wind tunnel tests performed by means of an active turbulence generator and a multibox deck sectional model. Finally, the linear and nonlinear approaches are compared with full scale simulations of the numerical response of a sectional bridge to an incoming real turbulent wind. & 2013 Elsevier Ltd. All rights reserved.

Keywords: Aerodynamic nonlinearities Time domain Rheological model Aerodynamic force transfer functions Numerical–experimental validation Multi-box deck section Band superposition approach

1. Introduction The synthesis of a numerical model able to simulate effectively the aerodynamic loads on a bridge deck section and their nonlinearities is still an open issue. In particular, in this work, we will investigate the dependence of aerodynamic forces on the reduced velocity Vn and on the fluctuations of the angle of attack a. A numerical approach to deal with such aerodynamic nonlinearities was proposed Diana et al. (2008). The approach, therein described, is based on the possibility to reproduce both the motion induced and the buffeting forces by means of a single rheological model that relies on the definition of an instantaneous angle of attack between deck and wind. This parameter, used as input to the numerical model, is made by a combination of the deck motion and turbulent wind velocity components. The rheological model computes the aerodynamic forces by summing up the forces that a group of simple mechanical systems (springs, dampers, coulomb friction elements and bump stops) would produce if they were moved or deformed according to the value of the instantaneous angle of attack. The parameters of the mechanical system that may be constant and/or variable are identified using specific wind tunnel tests that have to be performed on a sectional model in order to measure the aerodynamic

n

Corresponding author. Tel.: þ390223998360. E-mail address: [email protected] (T. Argentini).

0167-6105/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.jweia.2012.12.005

hysteresis loops, as defined in the cited paper. This rheological model can reproduce the nonlinear effects that arise in the selfexcited forces on a multi-box deck section, when different reduced velocities and amplitudes of variation of the angle of attack are considered for the rotational degree of freedom of the deck. In 2010 the same authors presented a validation of the proposed methodology applied to a single box deck section (the procedure is described in Diana et al., 2010). A different rheological model was proposed: springs and dampers have nonlinear characteristics, defined by a polynomial expression in order to make the identification of the coefficients easier, relying on the aerodynamic hysteresis loops measured in the wind tunnel. In that case, the rheological model was fully developed, considering both the aerodynamic forces produced by the torsional and vertical components of the motion (motion induced terms) and those produced by the turbulent wind vertical component (buffeting terms). In that context, it was shown how very similar nonlinear effects may be found if the three different situations are compared by means of aerodynamic hysteresis loops and a properly definition of the instantaneous angle of attack. Moreover, the numerical model was validated by comparing its results to wind tunnel experimental results obtained by exciting an elastically suspended deck sectional model through a highly correlated turbulent wind produced by an active turbulence generator able to produce harmonic vertical wind components (Diana et al., 2004). Such numerical approach showed the possibility to implement a numerical model

G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

able to simulate in the time domain the nonlinear dependence of the aerodynamic forces on the amplitude of the angle of attack and on the reduced velocity that are both present in the analysis of the experimental results. This approach that was successfully applied to study the aerodynamics of single box deck sections seems to be less appropriate when multi-box deck section is considered. During the aerodynamic design of the Messina bridge, aerodynamic hysteresis loops, produced by torsional and vertical motion components and produced both by turbulent wind velocity components, were measured at different reduced velocities and angles of attack, but it was not possible to identify a single input parameter leading these different situations to be aerodynamically comparable. In such cases, to analyze the nonlinear bridge response to turbulent wind, a different approach is proposed, and, in the present paper, a numerical–experimental investigation of the nonlinear aerodynamic effects of a multi-box deck section will be presented in order to propose an alternative way to deal with the problem. To this end, an improved version of the band superposition method has been developed, and validated against wind tunnel experimental results. The band superposition (BS) method has been already presented in literature by different authors (Diana et al., 1995; Chen and Kareem, 2001) and it relies on the assumption that nonlinear effects acting at low-frequency (LF band) are mainly induced by the large fluctuations of the instantaneous angle of attack, since the reduced velocity dependence is weak, while the nonlinear effects at high frequency (HF band) may be modeled by a linearized approach around the LF band solution, taking into account the dependence on both the reduced velocity and the angle of attack through a numerical model whose parameters change with time. The BS hypothesis originates, on one hand, from the evidence that, in full scale measurements of the wind approaching bridge decks, large fluctuations of the instantaneous angle of attack occur (Bocciolone et al., 1992). These fluctuations are mainly produced by the low frequency components as it can be seen in Fig. 1, where we report the time history of the low frequency angle of attack measured at Punta Faro, Messina, Italy on March 9th, 2010: values of 91 are present. On the other hand, looking at wind tunnel aerodynamic coefficients of multi-box deck section, a large dependence on the angle of attack is present (Diana et al., 2004).

Fig. 1. Messina Straits: full scale wind speed measurement of the instantaneous angle of attack at Punta Faro on March 9th, 2010 at 94 m above sea level. The mean horizontal wind speed is 23 m/s. The time history has been processed with a low-pass filter with a cut frequency of 0.05 Hz.

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A positive feature of band superposition approaches is that they rely only on the knowledge of flutter derivatives and aerodynamic admittance functions as a function of the reduced velocity and of the mean angle of attack. These are coefficients that are usually measured during the aerodynamic design of important bridges. However, standard tests provide these coefficients in a limited range of mean angles of attack (731, or sometimes 761), while a wider range is needed to apply band superposition methods. A wide range is needed because extrapolation of coefficients from existing databases is not acceptable due to expected nonlinear behavior at large angles of attack (see as an example Argentini et al., 2012; Diana et al., 2010). The number of bridges offering a complete dataset of aerodynamic coefficients tends to increase with the recent bridge design projects and the possibility to implement the band superposition method is a real opportunity, since no additional wind tunnel tests are required as it would happen for rheological models based on hysteresis loops. As a first step, in the present paper, the basic hypothesis of the BS approach is validated by means of specific wind tunnel tests. These tests are designed to highlight the effects of LF fluctuation of the angle of attack on high frequency forces. To this end, using a specific test rig developed at the wind tunnel of Politecnico di Milano, an highly correlated low frequency mono-harmonic fluctuation of the vertical component of the incoming wind velocity is generated. This LF fluctuation interacts with a dynamometric sectional model of a bridge with a multi-box crosssection. Besides the LF fluctuation, high frequency forces are generated either by an additional harmonic HF perturbation of the vertical component of the wind speed or by a HF harmonic motion of the model. Comparing the results of this kind of tests with those of standard tests (e.g. flutter derivatives around a static angle of attack), it is possible to verify that the large fluctuations of the instantaneous angle of attack are able to affect the high frequency force components. An improved version of the BS numerical model is therefore proposed to study the aerodynamic response of bridges with multi-box deck sections. The LF band contribution is obtained by considering the low frequency part of the wind spectrum and computing the deck response, through a corrected quasi-steady approach, in agreement with the previous version (Diana et al., 1995). The HF band contribution is computed through a linearized approach that uses the LF response as the slow varying solution around which the linearization is applied. The linear approach in the HF band is implemented in time domain in order to be able to consider the time variation of the LF band solution, and it takes into account, the dependence of the aerodynamic forces both on the reduced velocity and on the angle of attack. While the previous version of the numerical model adopted a decomposition of the high frequency range in several bands in order to follow the reduced velocity dependence of the aerodynamic forces, the new version exploits a rheological model to contemporary reproduce the reduced velocity and the angle of attack dependence in time domain on the whole HF band without asking for a HF band decomposition. The proposed rheological model operates in a similar way of approaches based on impulse response functions adopted by other authors (Chen and Kareem, 2001), and it exploits the information contained in the aerodynamic transfer functions (flutter derivatives and aerodynamic admittance functions), usually known in the frequency domain through wind tunnel tests at fixed mean angles of attack, and implemented in time domain. The developed rheological model, adopted for the HF band response, is a mechanical model that is similar to what was used by the present authors to deal with the nonlinear effects on single box deck sections. It is made by a group of mechanical elements whose response to a deformation, proportional to the fluctuation

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G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

of the angle of attack, produces a force that represents the aerodynamic load acting on the deck. The identification of the parameters of the single mechanical element in this case does not require the measure of aerodynamic hysteresis loops as it happens in the full nonlinear model, but it is based on the information contained in the flutter derivatives and admittance function coefficients measured at different mean angles of attack and different reduced velocity. With a single morphology of the mechanical system it is possible to reproduce the aerodynamic transfer function dependence on the reduced velocity and angle of attack, by using parameters of the rheological model, that are function of the angle of attack. The rheological model is therefore able to provide a continuous interpolation of the aerodynamic transfer function in the field where Vn and a were varied during flutter derivatives and admittance functions measurements. As a final step of the presented research activity, a numerical vs. experimental validation of the band superposition approach is proposed. The validation consists in comparing the high frequency aerodynamic buffeting and motion induced forces, acting on a dynamometric section model, excited by an additional low frequency turbulent wind actively generated in wind tunnel. The wind tunnel tests for the validation are properly designed and performed in order to highlight the non-linear effects occurring by the LF band and HF band components occurring in the real bridge operating conditions under laboratory controlled conditions. The validated nonlinear model is finally used to simulate the dynamic response of a bridge excited by an incoming turbulent wind with a full spectrum of harmonics and the results are compared with those obtained using a linear model for aerodynamic forces.

2. The band superposition approach 2.1. The BS hypothesis The Band Superposition (BS) approach was developed following the consideration that even if the bridge deck aerodynamics is dependent on the amplitude of the fluctuations of the angle of attack and on the reduced velocity, it is possible to separate the LF (low frequency) response of the bridge to turbulent wind from the HF (high frequency) response. The separation is feasible since the two regions show different features. The features of the LF response are as follows:

 The aerodynamic forces show a small dependence on the reduced



velocity, while they depend on the angle of attack. As an example to support this statement Fig. 5 reports the trend of the unsteady aerodynamic moment coefficient (flutter derivatives for torsional motion) as a function of the reduced velocity, for different mean angles of attack. Large variations of the instantaneous angle of attack are mainly related to the fluctuation of the wind velocity components, since small oscillations of the bridge are expected for stable aerodynamic solutions. These variations are caused by the large turbulent scales present in the atmospheric turbulent wind. The large turbulent structures that are able to generate fluctuations of the angle of attack that could be effective on the bridge deck are, in fact, those having large length scale (usually several times larger than the deck chord) and long time scale (low frequency).

 The fluctuation of the high reduced frequency wind velocity components are characterized by small turbulence length scales with respect to the deck chord length. These produce a flow field variation around the deck section that does not contribute to the global angle of attack. The previous considerations imply that, when large fluctuations of the angle of attack occur, in the LF range, a linear modeling of the aerodynamic forces is not appropriate in presence of strong nonlinear dependence of the forces on this parameter. In this case, a time domain approach, such as a quasi-steady theory (QST), that is able to consider the force dependence on large fluctuations of the angle of the attack but not on the reduced velocity is appropriate. In the following a corrected quasi-steady theory will be applied in order to take anyway into account that even at very low frequency the aerodynamic terms are slightly different from what is obtained using the aerodynamic static coefficients because of the rate of change of the angle of attack. On the other hand, in the HF range, when small fluctuations of the angle of attack are produced by the wind velocity fluctuations, the aerodynamic forces should be subjected only to the reduced velocity dependency, if they were dependent just on the constant mean angle of attack. The basic hypothesis of the BS approach is that the mean angle of attack, perceived by the deck in the definition of the aerodynamic force components acting in the HF range, is not only the static mean value but it is also related to the fluctuation that happens at low frequency with high amplitude (perceived as static region for the HF terms). Therefore the method suggests that the HF aerodynamic forces are related to the instantaneous angle of attack computed considering the LF deck response and wind spectrum and not to the static mean angle of attack. If we now consider that the fluctuation of the instantaneous angle of attack in the HF range are small, this means that the problem may be modeled using a linearization of the aerodynamic forces evaluated around the LF instantaneous angle of attack. Under this assumption, it is possible to model the aerodynamic forces in the HF range by means of the aerodynamic transfer function that is commonly expressed in bridge aerodynamics through the flutter derivatives coefficients, usually measured by wind tunnel tests at different reduced velocities and mean angles of attack. In this case, being the linear problem considered around a slowly varying solution, represented by the low frequency fluctuation of the angle of attack, the response of the bridge deck is modeled applying, at each time step, the aerodynamic transfer function correspondent to the instantaneous value of the LF angle of attack. If the BS hypothesis held, the bridge deck response to turbulent wind, might be performed by dividing the problem in two parts: a nonlinear problem at low frequency modeled using a corrected quasi-steady approach, and a nonlinear problem at high frequency modeled using a modulation of the aerodynamic transfer functions performed on the basis of the LF fluctuation of the angle of attack. 2.2. Experimental tests To validate this assumption, specific wind tunnel forced motion tests were designed and carried out using a dynamometric deck sectional model. Tests are aimed to compare two situations reproduced by the following two test procedures:

The features of the HF response are as follows:

 Standard forced motion type test: to measure flutter derivatives  The aerodynamic forces show large dependence both on the reduced velocity and on the mean angle of attack.

and aerodynamic admittance functions around a static mean angle of attack (linear behavior around a fixed position).

G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

 Band Superposition type test: to measure harmonic high frequency self-induced and buffeting forces around a sinusoidal slowly varying angle of attack with large amplitude (linear behavior around a slowly varying solution). In such tests the low frequency fluctuation is present in the incoming turbulence. The aerodynamic forces acting on the dynamometric section of the model are measured in both situations and the comparison of the higher frequency terms will be presented and discussed in the following. 2.2.1. Setup Wind tunnel tests were performed at the Politecnico di Milano wind tunnel, in the boundary layer tests section (http://www. windtunnel.polimi.it) on a deck sectional model representing a bridge multi-box deck shape analyzed during the Messina Bridge design stage. The deck section, sketched in Fig. 2, is made by three boxes connected by transversal beams, and it is characterized by tall wind screens positioned at the leeward edges. The model is 4 m long, has a chord B of 1.33 m and a central dynamometric part 1.33 m long. The measurement of the flutter derivatives coefficients at different mean angles of attack and different reduced velocities is performed using the forced motion test procedure described in Diana et al. (2004). The dynamometric deck sectional model is moved by three hydraulic actuators (see Fig. 3). The deck motion is controlled in order to keep constant motion amplitudes, corresponding to small equivalent variations of the angle of attack (about 711), at a fixed frequency.

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The test rig is completed by an active turbulence generator made by an array of horizontal airfoils driven to harmonically rotate around their axis in order to produce a widely spanwise correlated perturbation of the flow at a specific frequency. The active turbulence generator is positioned six deck chords upstream the model (Fig. 3). Fig. 4 reports pictures of a smoke visualization of the sinusoidal wave produced by the active turbulence generator. The turbulent wind velocity components are measured by a 4-hole pressure probe positioned one chord upwind the model at the same height of the deck.

2.2.2. Force measurement The selected multi-box deck section is well suited to study the aerodynamic force dependence on the angle of attack and on the reduced velocity since it presents both the aspects. As an example Fig. 5 shows the an2 and an3 flutter derivatives coefficients trend on the reduced velocity parameter V n ¼ V=ðfBÞ, for different mean angles of attack. Flutter derivatives coefficients are reported according to the Zasso (1996) formulation, which defines the self excited part of the aerodynamic forces per unit length as: ! z_ 1 By_ p z n y_ p y n þ p p þ pn3 y þpn4 Dse ¼ rV 2 B pn1 pn2 5 6 V V 2 V 2V no2 B 2V no2 B ð1Þ

Lse ¼

z_ y_ 1 By_ p z p y rV 2 B hn1 hn2 þ hn3 y þ hn4 n2 hn5 þ hn6 n2 V V 2 V 2V o B 2V o B

Fig. 2. Messina bridge multi-box deck section.

Fig. 3. Wind tunnel test rig.

!

ð2Þ

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G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

Fig. 4. Smoke visualization of the sinusoidal wave.

Fig. 5. an2 (a) and an3 (b) vs. Vn for nine different mean angles of attack a , according to Eq. (3).

z_ 1 By_ p z n y_ p y a5 þan6 þan3 y þ an4 M se ¼ rV 2 B2 an1 an2 V 2 V V 2V no2 B 2V no2 B

!

ð3Þ where r is the air density, V is the mean wind speed, B is the deck chord, o is the circular frequency of motion, y, z, y are respectively

the lateral displacement, the vertical displacement, and the rotation n of the deck. an ,h ,pn are the flutter derivative coefficients, function of the reduced velocity V n ¼ V=fB ¼ ðV=oBÞð1=2pÞ ¼ V no =ð2pÞ and mean angle of attack a . Sign conventions are reported in Fig. 9. Considering the buffeting forces per unit length, the aerodynamic admittance functions are reported according to the

G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

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following formulation: Dbuff ¼

1 w rV 2 BðwDw Þ 2 V

ð4Þ

Lbuff ¼

1 w rV 2 BðwLw Þ 2 V

ð5Þ

M buff ¼

1 w rV 2 B2 ðwMw Þ 2 V

ð6Þ

where w is the harmonic vertical incoming turbulence, wn are the complex aerodynamic admittances, function of Vn and a . They are reported in Figs. 17–19. During the band superposition type tests, a large fluctuation of the angle of attack at high reduced velocity (representing the effect of the large turbulence scale of the wind) is produced with the active turbulence generator. The small fluctuation of the angle of attack at low reduced velocity can be either a high frequency motion of the deck or an additional high frequency component of the vertical wind velocity (representing the linear part of the BS approach). Figs. 6 and 7 show respectively the time histories of two tests with the same LF variation of the angle of attack at 0.25 Hz ðV n ¼ 35Þ with amplitude 751 and with HF components at 2.4 Hz ðV n ¼ 3:7Þ with amplitude 711. In the first figure the HF component is due to a deck rotation y, while in the second one it is due to an additional harmonic contribution in the vertical component of wind velocity. Together with the angle of attack, the time histories of the measured aerodynamic forces are reported to highlight the strong dependence of HF components on the LF fluctuation of a. In particular, it can be noticed that in both situations the amplitude of the high frequency components of the drag (D) and the moment (M) strongly depend on the instantaneous angle

Fig. 7. Time histories of buffeting forces considering an instantaneous angle of attack with large LF fluctuation of w plus a small HF vertical turbulence w.

of attack. On the contrary, the amplitude of the lift (L) component seems less dependent of the LF fluctuations. If we compare this behavior with the trend of flutter derivatives and admittance functions at V n ¼ 3:7 (e.g. Fig. 5), we notice that the coefficients at this reduced velocity have a strong nonlinear dependence on the angle of attack which reflects the experimental evidence in band superposition tests. As an example, if we analyze the unsteady moment due to buffeting, we see from Fig. 7 that HF force oscillations are higher when the LF angle of attack is negative: this is coherent with the magnitude of wMw at V n ¼ 3:7 reported in Fig. 19a that varies from 0.1 at a ¼ 61 up to 0.5 when a ¼ 61. On the contrary, the lift coefficient wLw (Fig. 18a) has smaller variations of magnitude (1.7–2.4) that are only slightly modulated by the low frequency angle of attack in the band superposition test. Similar comments can be done for the other force components. The results demonstrate how the low frequency variation of the angle of attack may influence the aerodynamic behavior at higher frequency in situations similar to the analyzed one where a large dependence of the unsteady force coefficients on the angle of attack is present. In the following the dependency of the buffeting terms acting at high frequency on the LF fluctuation of the aerodynamic admittance functions will be taken as experimental reference case for the validation of the BS numerical approach and will be there discussed.

3. A new band superposition numerical model

Fig. 6. Time histories of motion induced forces considering an instantaneous angle of attack with large LF fluctuation of w plus a small HF rotation y.

Starting from the experimental evidence of the validity of the BS approach, a new implementation of the methodology is here proposed. In a previous version of the BS method (Diana et al.,

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G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

Step 1

LF turbulent wind time history Step 2 Static coefficients

Step 3

Nonlinear corrected QST

LF angle of attack α B1

Flutter derivatives

Rheological model for aeroelastic effects HF response LF response

Total response Fig. 8. Band superposition approach: simulation flowchart.

1995) the dependency of aerodynamic forces on Vn in the HF range was computed by dividing the HF range into several bands, assuming the aerodynamic coefficients to be Vn independent within each band. At each Vn band a corresponding wind time history was selected using a band pass filter. The bridge response to each HF band is then computed and summed to all the others in order to have the full HF response. This methodology has some drawbacks, due to the large number of bands that have to be considered if unsteady force coefficients show large gradients with Vn or the need to restrict the bridge response computation to just the structural modes within or close to the Vn band. Other researchers proposed alternative methods to overcome these limitations, relying on either neural networks (Wu and Kareem, 2011) or rational function approximations (Chen et al., 2000; Chen and Kareem, 2001, 2003) that are able to consider the whole HF response in a single band. In the present paper a rheological model approach is adopted to compute the bridge response in the HF range without splitting into bands. The algorithm for the simulation of the nonlinear response to turbulent wind is schematically reported in Fig. 8 and

described in details in the following paragraphs. In particular the procedure consists of three main steps: 1. LF–HF range threshold definition 2. LF response computation 3. HF response computation

3.1. Step 1: LF–HF range threshold definition The threshold separating the LF range from the HF range has to be defined in terms of reduced velocity Vn or reduced frequency fn. The threshold delimits the region, at high reduced velocity, where the flutter derivatives coefficients and the aerodynamic admittance function coefficients show a small dependence on the reduced velocity. Once the Vn threshold is defined, it can be used to separate the wind spectrum into two bands. From the analysis of the Messina unsteady aerodynamic coefficients the threshold may be set at V n ¼ 15. Usually, a time–space reconstruction of the wind field is performed to compute the bridge response to turbulent wind,

G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

exploiting the knowledge of the turbulence statistical properties of the wind at the site where the bridge will be built. Several numerical methods have been proposed by different authors (e.g. Deodatis, 1996; Ding et al., 2006) in order generate wind time histories in different points, distributed along the bridge elements, taking into account the space correlation among the different wind velocity components at the different locations and fulfilling the statistical properties that characterize the target turbulent wind. Once the time history at a specific point is generated, it is possible to separate the LF part from the HF part by using a low-pass filter. As far as the lack of reduced velocity dependence of the aerodynamic coefficients in the LF range allows for the adoption of a quasi-steady nonlinear approach, it is possible to apply a linear approach in the HF range since the single harmonic components does not produce large variations of HF angle of attack. 3.2. Step 2: LF response computation The LF response computation is performed by a corrected quasi-steady approach (Diana et al., 1995). According to the QST, aerodynamic drag, lift and moment, acting on the deck section per unit length, are DðtÞ ¼ 12

2

rB½V rel ðtÞ C D ðaLF,D ðtÞÞ

ð7Þ

LðtÞ ¼ 12rB½V rel ðtÞ2 C L ðaLF,L ðtÞÞ

ð8Þ

MðtÞ ¼ 12rB2 ½V rel ðtÞ2 C M ðaLF,M ðtÞÞ

ð9Þ

where C D,L,M are the static aerodynamic coefficients, r is air density, B is the deck chord, Vrel(t) is the instantaneous relative wind speed , aLF ðtÞ is the instantaneous angle of attack. The static force coefficients and the corresponding D, L, and M sign conventions are shown in Fig. 9. The angle of attack aLF,j has different subscripts ðj ¼ D,L,MÞ for each force component because the corrected QST requires that quasi-steady aerodynamic effects are taken into account in the

47

definition of the angle of attack. aLF ðtÞ is defined analogous to standard QST as 0

aLF,j ¼ yLF þatanB @

_ LF 1 w wLF z_ LF þ B1,j y_ LF þB2,j V C A V þ vLF y_ LF

ð10Þ

where yLF is the deck rotation, y_ LF , z_ LF , and y_ LF are respectively the deck velocities of rotation, vertical, and horizontal movement; wLF and vLF are thevertical and horizontal components of the incoming turbulent wind velocity respectively. B1,j and B2,j aredimensional coefficients that are introduced to correct the QST . The correction introduced by Diana et al. (1995) consists in selecting the B1,j length from the flutter derivative coefficients as B1D ¼

pn2 B; p3 C L n

n

B1L ¼

h2 B; h3 þ C D n

B1M ¼

an2 B an3

ð11Þ

evaluating the flutter derivatives at high Vn ð 4 15Þ, where they are almost constant. B2,j is here introducedto account for time delays between incoming LF turbulent wind and LF aerodynamic forces. B2,j is evaluated from aerodynamic admittance functions at high Vn as B2D ¼

IðwDw Þ V n B; RðwDw Þ 2p

B2L ¼

IðwLw Þ V n B; RðwLw Þ 2p

B2M ¼

IðwMw Þ V n B RðwMw Þ 2p

ð12Þ

where I and R indicate the imaginary and the real part of the admittance functions respectively. aLF,D , aLF,L , aLF,M are in general different. Flutter derivatives and aerodynamic admittances represent a transfer function between the aerodynamic forces acting on the deck and an input variable the is either a deck harmonic displacement or incoming wind harmonic component. At very high Vn these transfer functions approach the linearized QST values. However, experimental evidence (e.g. ? Diana et al., 2008) highlighted that a small phase lag between force and input is present at the typical Vn of LF band, and standard QST does not model it correctly.

Fig. 9. Static aerodynamic force coefficients and the corresponding sign conventions.

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G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

The phase lag Dj can be accounted for by identifying B1,j and B2,j from the aerodynamic transfer functions, considering as input y and w. As an example for the aerodynamic moment, B1,M and B2,M are related to the phase lags DjM, y and DjM,w by the following expressions: B1M ¼ tanðDjM, y Þ

Vn B 2p

ð13Þ

3.3. Step 3: HF response computation Unsteady HF aerodynamic forces are modeled in the time domain with a rheological model whose parameters depend on the LF instantaneous angle of attack. Self-excited and buffeting forces are modeled independently and their effects are summed up exploiting the superposition hypothesis, i.e.: _ HF , v_ HF Þ F unsteady ðtÞ ¼ F se ðaLF ðtÞ,xHF , x_ HF , x€ HF Þ þ F buff ðaLF ðtÞ,wHF ,vHF , w ð15Þ

n

B2M ¼ tanðDjM,w Þ

V B 2p

ð14Þ

moving base

Fig. 10. Scheme for rheological model.

Eq. (15) highlights that unsteady forces depend explicitly on the LF angle of attack and on HF deck motion and HF incoming turbulence. Each force component is modeled with a specific rheological model which consists in several independent mechanical oscillators of order 2, 1, and 0 placed in parallel configuration (see Fig. 10). The unsteady nonlinear aerodynamic force acting on the deck is equal to the force necessary to impose a motion c to the base of the system . The input c is directly related to the HF deck motion and incoming turbulence; the mechanical parameters of the model (mass, stiffness, damping) depend on the instantaneous LF angle of attack and they are identified using the unsteady force coefficients at static mean angles of attack. The identification of the rheological model pass through the concept of aerodynamic force transfer function. As an example, analyzing the self excited aerodynamic moment induced by a torsional motion of the deck, the force component per unit length

Fig. 11. Experimental and identified numerical aerodynamic transfer function T nMy vs. fn in terms of magnitude and phase, for several mean angles of attack.

G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

can be expressed as   1 2p 1 n M se ¼ rV 2 B2 an3 i n an2 yHF ¼ rV 2 B2 ½T nMy ðast ,f ÞyHF 2 2 V

ð16Þ

where T nMy is the aerodynamic transfer function between the aerodynamic moment and the input c ¼ yHF . T nMy is a complex

Fig. 12. Parameters of the rheological model as a function of the mean angle of attack.

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function of the reduced frequency fn and of the static angle of attack a . Fig. 11 shows its magnitude and phase as a function of fn for several angles of attack. The procedure to identify the parameters of the model consists in selecting the number and order of oscillators for a ¼ 01 and the value of the parameters through a constrained optimization that imposes the dynamic stability of the subsystems. The values for the other a are found imposing that their value can change at a maximum rate of 10%/1 starting from those obtained for a ¼ 01. For unknown angles, aerodynamic transfer functions are interpolated linearly. Fig. 11 shows the identified aerodynamic transfer functions for the moment component, obtained using a rheological model consisting in two second order and one first order oscillators. Fig. 12 shows the trend with the angle of attack of the parameters of these oscillators. Similar results hold for the other eight selfinduced terms and for the three admittance terms. Once all the rheological models are identified, the HF response of the system is computed simulating separately self-excited and buffeting terms. The buffeting force is computed integrating numerically the dynamic equations of the rheological models, whose parameters are modulated by the instantaneous LF angle of attack. The result of the integration that represents the nonlinear time history of the buffeting force is fed to equation of motion of the bridge deck. The self-excited terms, instead,

Fig. 13. Experimental and numerical spectra of BS test wLF þ yHF : (a) a, (b) experimental aerodynamic forces, (c) results of the linear approach, (d) results of the BS approach.

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G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

are integrated together with the equations of motion of the deck, using a state space formulation. The total response is finally computed as the sum of LF and HF responses.

4. Experimental and numerical results As a first step of experimental–numerical validation, in Section 4.1 we start analyzing the results obtained by the linear and nonlinear models concerning the HF self-excited aerodynamic forces induced by a torsional motion ðyHF Þ in presence of a variation of the instantaneous angle of attack induced by a LF fluctuation of the w wind component (wLF). Then, in Section 4.2, we move to the analysis concerning the HF buffeting forces (wHF) in presence of a LF variation of the instantaneous angle of attack induced by a LF fluctuation of the w wind component (wLF). Finally, in Section 4.3, we simulate the response of a 3-dof model of a bridge deck section excited by a turbulent wind with a complete spectrum of harmonics, for three different mean wind velocities, comparing the results of the nonlinear rheological model with those of a linear one.

Table 3 Comparison on the Mx force component for BS test wLF þ yHF . Frequency (Hz)

0.25

0.5

2.15

2.4

2.65

Mx, Exp (mod) (phase) Mx, Lin (mod) (phase) Mx, BS (mod) (phase)

1.78 Nm  231 1.84 Nm  281 1.7 Nm  271

0.38 Nm 2241 – – 0.43 Nm 2161

0.27 Nm 2171 – – 0.26 Nm 2181

0.80 Nm  651 0.7 Nm  741 0.68 Nm  681

0.39 Nm 651 – – 0.34 Nm 691

In the following paragraphs, for each force component experimental, linear, and BS results are analyzed in detail in order to evaluate how the motion induced aerodynamic terms are modeled by the different numerical approaches.

4.1.1. Drag force Linear approach: Analyzing the results of the linear approach, the following comments can be done for the HF harmonics:

 A single harmonic contribution is modeled by the linear model

4.1. Band superposition test: wLF þ yHF Fig. 13 compares the spectra of the aerodynamic forces and moments, acting on the sectional model, while it is rotating around its axis at 01 mean angle of attack with an harmonic motion yHF at high frequency (2.4 Hz) and it is run over by a turbulent flow with a single vertical velocity component at 0.25 Hz with large amplitude (wLF). The equivalent fluctuation of the instantaneous angle of attack, produced by the low frequency component, present in the incoming wind, is equal to n 51 (acting at V n ¼ 35 or f ¼ 0:03), while the one produced by the deck motion at high frequency is five times lower (1.21 at V n ¼ 3:7 n or equivalently f ¼ 0:27). The mean wind speed is 11.9 m/s. In Tables 1–3 the modulus (mod) and the phase of the most important harmonics are compared. Phase shifts for the 0.25 and 0.5 Hz force components are relative to the 0.25 Hz phase of a, while phase shifts for the HF force components are relative to the 2.4 Hz phase of a.



Moving to the LF force components, we notice that:

 The 0.25 Hz harmonic has a small phase shift with respect to the

Table 1 Comparison on the Fy force component for BS test wLF þ yHF . Frequency (Hz)

0.25

0.5

2.15

2.4

2.65

Fy, Exp (mod) (phase) Fy, Lin (mod) (phase) Fy, BS (mod) (phase)

2.24 N  81 2.47 N 1191 3.57 N  151

1.56 N  911 – – 1.5 N  1051

1.03 N  2901 – – 0.99 N  3071

1.6 N  3431 1.38 N  3101 1.5 N  3421

1.19 N  891 – – 0.97 N  671

Table 2 Comparison on the Fz force component for BS test wLF þ yHF . Frequency (Hz)

0.25

0.5

2.15

2.4

2.65

Fz, Exp (mod) (phase) Fz, Lin (mod) (phase) Fz, BS (mod) (phase)

12.26 N  31 7.3 N  31 10.1 N  41

– – – – – –

– – – – – –

8.65 N 191 7.54 N 261 8.2 N 191

– – – – – –

at high frequency (2.4 Hz), while additional terms are present in the experimental one. More in detail, a sub-harmonic (at 2.15 Hz) and a super-harmonic (at 2.65 Hz) appear in the experimental spectrum. These harmonics correspond to a nonlinear interaction between the slowing varying fluctuation that is present in the wind and the higher frequency due to the deck motion. The two frequencies represent, in fact, respectively the difference and the sum of 2.4 Hz and 0.25 Hz. Since these terms are not present in the wind or in the controlled deck motion the linear approach is not able to reproduce them. The modulus of the 2.4 Hz term is slightly underestimated (  14%) and a discrepancy of about 301 is present in the phase between the force and the torsional motion y.



corresponding harmonic term of the incoming wind in the experimental spectrum, while the linear approach simulates a force with a very large phase shift. This discrepancy is due to the critical measure of the phase of the aerodynamic admittance function for the drag component at the considered angle of attack (equal to 01). At 01, the aerodynamic transfer function is small since it is related to the slope of the static drag coefficient at 01 that is close to the minimum of the function (see Fig. 9). The drag coefficient is an almost quadratic function of the angle of attack, presenting positive slope at positive angles of attack and negative slope at negative angles of attack and a minimum that not necessarily correspond to 01. At the very high reduced velocity of the LF term, this means that the aerodynamic transfer function may present either negligible phase shift on the positive angles of attack or it is out of phase at negative angles of attack (as can be appreciated by the trend of the phase of the admittance function wDw reported in Fig. 19). The linear model keeps the phase shift measured at 01 constant for the LF term. For the present case, this phase shift is similar to what measured at the positive angles of attack according to a sort of interpolation between the behavior at positive and negative angles of attack on the basis that for the considered deck, the minimum of the drag curve occurs at a slightly negative angle of attack. The experimental component at 0.5 Hz, acting at twice the frequency of the LF component of the incoming wind has a magnitude that is comparable to the 0.25 Hz one and it can not be represented by the linear approach.

G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

51

Fig. 14. Experimental aerodynamic transfer function T nDy vs. fn in terms of magnitude and phase, for several mean angles of attack.

BS approach: As far as the HF force is concerned, the BS approach has the following performances:

 Phase shifts of LF components are better reproduced, on the

 It is able to reproduce both the sub-harmonic (2.15 Hz) and the





super-harmonic (2.65 Hz) terms. In this nonlinear approach the aerodynamic coefficients to compute the motion induced aerodynamic forces are not evaluated at the mean angle of attack as it happens in the linear approach but at the instantaneous angle of attack produced by the fluctuation of the incoming flow. Being the flutter derivatives coefficients dependent by the angle of attack, as highlighted in Fig. 14 where the aerodynamic transfer functions T nDy are reported vs. fn for different mean angles of attack, the aerodynamic force, computed in the BS approach is modulated in amplitude at the same frequency of the wind fluctuation (0.25 Hz). The modulation of the aerodynamic coefficients on a dynamic force acting at 2.4 Hz, as expected from the Werner trigonometric formula, produces the sub and super harmonics terms. The modulus of the 2.4 Hz term is underestimated (  6%) compared to the experimental one even if the discrepancy is less than the half of the linear approach one. A very good agreement is furthermore found on the phase shift. The moduli of the sub- and super-harmonics are well reproduced by the BS approach (respectively  4% and  18%) while a discrepancy of almost 201 is present on their phase shifts.

Analyzing the LF force components, we notice that:



contrary of what happened with the linear approach. In this case the phase shift of the aerodynamic force is due to the value of the B2D coefficient, at least for the 0.25 Hz term. Even though the B2D coefficient for the drag force is computed according to the phase of the corresponding admittance function, in the nonlinear BS approach this value is not taken at a constant mean angle of attack but it follows the slowly varying instantaneous angle of attack. During the fluctuation of the instantaneous angle of attack the parameter B2D may assume both positive and negative values according to the possibility to experience positive and negative instantaneous angles of attack. The value of the phase delay of the first low frequency harmonic is therefore due to the values assumed by the B2D parameter during the complete fluctuation of the angle of attack. It is more difficult to comment on the phase shift of the harmonic at 0.5 Hz whose presence is due to the quadratic trend of the drag static coefficient with a that is followed during the fluctuation of the instantaneous angle of attack in the corrected quasi-steady approach.

4.1.2. Lift force Linear approach: Concerning the experimental and linear approach force spectra of the lift, they look like very similar and just the two harmonics at 0.25 Hz and at 2.4 Hz are present. Considering the HF term, the following discrepancies are present:

 The BS results present both the harmonics at low frequency, the former at 0.25 Hz has an overestimated magnitude while the amplitude of the second one is well predicted.

 The linear model underestimates (  13%) the magnitude of the HF component.

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G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

 The linear model predicts a phase shift 71 smaller than the experimental one.

Considering the LF term, the linear model underestimates the magnitude of the lift force component acting at 0.25 Hz that is anyway acting in-phase with the corresponding wind contribution as it occurs in the experimental measurement. BS approach: Also the BS approach predict just two predominant peaks at 0.25 Hz and 2.4 Hz. The BS approach is able to better approximate the experimental results at 2.4 Hz both in terms of modulus (  5%) and phase shift. An explanation is possible observing the T nLy coefficient dependency on the angle n of attack at V n ¼ 3:7 ðf ¼ 0:27Þ (see Fig. 15). With respect to the linear approach that adopts the coefficient at 01 to compute the aerodynamic force the BS approach computes the force considering the aerodynamic coefficient at the instantaneous angle of attack. In Fig. 15 it is visible how the coefficient at 01 at that Vnis the lowest, compared to all the other angles of attack. During a period of fluctuation of the instantaneous angle of attack the value of the coefficient will be most of the time higher than the value assumed at 01 and this is reflected in the modulus of the aerodynamic force representing a sort of average of the force value on the whole period. Similar considerations hold for the phase shift. The 0.25 Hz contribution presents a magnitude that is higher than the one obtained through the linear approach and it is more similar to the experimental one. The reason is due to the different slopes of the lift static coefficient close to a ¼ 01 (Fig. 9). Also from

the analysis of the modulus of the aerodynamic admittance function for the lift force at high reduced velocity (using the Zasso formulation these values are asymptotic to the values of the slope of the static lift coefficient @C L =@a, see Fig. 18), it is evident that, on the negative angles of attack, the slope of the curve is higher if compared to what occurs at the positive angles of attack. The simulation of the LF band through the corrected quasi-steady approach takes into account this change in the slope of the lift curve since the lift force is computed considering the value of the aerodynamic coefficient at the instantaneous angle of attack, which presents fluctuations up to 51. The phase shift is similar to the experimental one and it is also similar to the linear approach one since the B2L coefficient does not present a wide dependency on the angle of attack.

4.1.3. Moment Linear approach: Sub- and super-harmonics are present in the experimental spectrum of the aerodynamic moment around the 2.4 Hz contribution, in a similar way of what happens in the drag force:

 The magnitude of the HF contribution at 2.4 Hz is underestimated (  14%) by the linear model that is anyway able to predict a similar value of phase shift due to the dependency of the buffeting force on reduced velocity as shown by the aerodynamic transfer function for the aerodynamic moment reported in Fig. 11 in terms of magnitude and phase for different mean angles of attack.

Fig. 15. Experimental aerodynamic transfer function T nLy vs. fn in terms of magnitude and phase, for several mean angles of attack.

G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

53

Fig. 16. Experimental and numerical spectra of BS test wLF þ wHF: (a) a, (b) experimental aerodynamic forces, (c) results of the linear approach, (d) results of the BS approach.

In the LF range, apart from the lack of the additional harmonics (0.5 Hz), the comparison between the experimental and the linear model results highlights that the linear model predicts both the magnitude and the phase delay of the LF component with a satisfactory degree of accuracy. BS approach: In a similar way of what was already commented on the drag force, the spectrum of the aerodynamic moment computed by the BS approach is characterized by the growth of additional harmonic contributions at 2.15 Hz, 2.65 Hz, and 0.5 Hz. We can comment that:

 The nonlinear variation of the slope of the static coefficient is also the reason for the genesis of the 0.5 Hz term. In this case, the moment coefficient trend vs. the angle of attack is not a simple quadratic curve (as in the drag force coefficient) but it is more complex. Furthermore the high performance of the Messina deck section design that is characterized by very low absolute values and slope of the moment coefficient, makes the correct simulation of this nonlinear effect more affected by uncertainties of the measurement of the aerodynamic coefficients. The result is anyway satisfactory and it anyway confirms the need for a nonlinear approach.

 The 2.15 Hz and 2.65 Hz contributions are very well captured



in the BS approach both in terms of magnitude (3% and  13% respectively) and phase shift. Also in this case, the considerations on the aerodynamic coefficient dependency on the angle of attack hold. On the contrary of what performed by the linear approach, the BS approach predicts the 0.25 Hz contribution with a similar phase shift but with underestimated magnitude. The lower magnitude is due to the reduction of the slope of the moment static coefficient moving from 01 towards the negative angles of attack, as visible in Fig. 9 and in the value assumed by the admittance function at high reduced velocity reported in Fig. 19. The phase shift of 281 is reproduced thanks to the B2M coefficient.

4.2. Band superposition test: wLF þwHF In Fig. 16 we compare the spectra of the aerodynamic forces and moments, acting on the sectional model, while it is kept at rest and it is run over by a turbulent flow, whose vertical velocity component is produced by the superposition of a 0.25 Hz component with high amplitude (wLF) and a 2.4 Hz component at lower amplitude (wHF), with zero phase shift. The equivalent fluctuation of the instantaneous angle of attack, produced by the LF component is equal to 5.31 (acting at V n ¼ 35), while the one produced by the HF component is five times lower (1.11 at V n ¼ 3:8). For sake of comparison, as already done in the previous test case, the

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G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

modulus (mod) and the phase of the most important harmonics, present in the spectra are reported in Tables 4–6. The comments for the LF components have been already provided in the previous paragraph, and in this section we will focus only on the analysis of the HF terms. 4.2.1. Drag force Linear approach: If we start comparing the experimental results with those obtained by the linear approach, the following considerations immediately arise:

 Additional harmonic components are present in the experi-





mental spectrum of the drag force beside the term at 2.4 Hz. The components at 2.15 Hz and 2.65 Hz arise from the combination of the two frequencies components of the incoming wind (LF and HF terms) corresponding respectively to their difference and their sum, both having a magnitude almost the half of the 2.4 Hz term and they act with a relevant phase delay with respect to the HF term. All these HF contributions are not present in the spectrum of the drag, obtained from the linear approach, since there are no harmonic contributions at those frequencies in the wind spectrum. The 2.4 Hz component magnitude and phase are underestimated by the linear model.

BS approach: Comparing the spectra of the results obtained by using the BS approach with the experimental results, it is possible to appreciate how most of the nonlinear effects that are not reproduced by the linear model are present. In particular:



closer to the experimental one. Looking at the admittance function dependence on the angle of attack reported in Fig. 17, n for V n ¼ 3:8 ðf ¼ 0:27Þ, it is possible to understand how, varying the instantaneous angle of attack, the magnitude of the force contribution increases because all the coefficient shows higher values at all the angles of attack at that Vn, if compared to the corresponding 01, and the contribution of the negative angles of attack on the phase shift operates in order to increase the delay. Concerning the two contributions at 2.15 Hz and 2.65 Hz, the BS method demonstrates to be able to simulate these terms in a satisfactory way, being the magnitude of the two harmonics equal and slightly underestimated with respect to the experimental ones and having a phase shift close to the experimental one.

4.2.2. Lift force Linear approach: The analysis of the experimental and linear lift force shows that they are very similar and that only the two harmonics at 0.25 Hz and at 2.4 Hz are present. The linear model underestimates the magnitude of the HF component ðV n ¼ 3:8Þ being anyway able to predict the large phase shift that is due the dependency of the buffeting force on reduced velocity, as shown by the aerodynamic admittance function for the lift force reported in Fig. 18 in terms of magnitude and phase for different mean angles of attack.

 Compared to the linear approach, the BS approach predicts a 2.4 Hz component with a higher magnitude and a phase shift Table 4 Comparison on the Fy force component for BS test wLF þ wHF. Frequency (Hz)

0.25

0.5

2.15

2.4

2.65

Fy, Exp (mod) (phase) Fy, Lin (mod) (phase) Fy, BS (mod) (phase)

1.64 N  21 2.56 N  2261 2.34 N  71

1.84 N 181 – – 1.8 N 231

1N  2101 – – 0.8 N  2201

1.71 N  1151 1N  441 1.46 N  1131

1N  1381 – – 0.8 N  1271

Table 5 Comparison on the Fz force component for BS test wLF þ wHF. Frequency (Hz)

0.25

0.5

2.15

2.4

2.65

Fz, Exp (mod) (phase) Fz, Lin (mod) (phase) Fz, BS (mod) (phase)

12.65 N  41 8.6 N  31 12.8 N  41

– – – – – –

– – – – – –

7.75 N  1221 5.95 N  1071 5.92 N  1291

– – – – – –

Table 6 Comparison on the Mx force component for BS test wLF þ wHF. Frequency (Hz)

0.25

0.5

2.15

2.4

2.65

Mx, Exp (mod) (phase) Mx, Lin (mod) (phase) Mx, BS (mod) (phase)

2Nm  211 2.17 N m  281 1.8 N m  251

0.36 N m  20.91 – – 0.43 N m  181

0.37 N m  2781 – – 0.31 N m  2971

1.03 N m  431 0.3 N m  251 0.9 N m  561

0.5 N m  2281 – – 0.62 N m  2391

Fig. 17. wDw vs. Vn for different mean angles of attack: magnitude (a) and phase (b).

G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

Fig. 18. wLw vs. Vn for different mean angles of attack: magnitude (a) and phase (b).

55

Fig. 19. wMw vs. Vn for different mean angles of attack: magnitude (a) and phase (b).

BS approach: In this case, since the unsteady lift force is not strongly dependent on a we get the following results:

 No significant additional harmonic contributions appears in the 

BS spectrum of the lift force. This is coherent with the experimental results. BS result is similar to the linear one for the 2.4 Hz contribution. If we look at the modulus of the admittance function reported in Fig. 18a, at V n ¼ 3:8, it is possible to see how the dependence on the angle of attack is very small, since we are close to point where all the curves intersect each other. Also the value of the phase shift is not largely affected by the angle of attack.

4.2.3. Moment Linear approach: The magnitude of the HF contribution at 2.4 Hz is largely underestimated by the linear model that is anyway able to predict a phase shift close to the experimental one. This phase shift is due to the dependency of the buffeting force on reduced velocity, as shown by the aerodynamic admittance function for the aerodynamic moment reported in Fig. 19 in terms of magnitude and phase for different mean angles of attack. In a similar way of what was already commented on the drag force, sub- and super-harmonics are present in the experimental result, and they respectively have a magnitude equal to the 50% and 40% of the 2.4 Hz component. These components are not reproduced by the linear model.

BS approach: For the nonlinear approach, the following comments can be done:

 A big difference between the linear and the BS approach is



found comparing the magnitude of the HF contribution at 2.4 Hz. This term that was largely underestimated by the linear model, is now well predicted both in magnitude and in phase shift. The quality of the result is again due to the dependency of the buffeting force on the angle of attack and on reduced velocity as shown by the aerodynamic admittance function for the aerodynamic moment reported in Fig. 19. In this case ðV n ¼ 3:8Þ the contribution of the negative angle of attack results in increasing the value of the moment magnitude on the contrary of what they induce at high reduced velocities. A better results is also obtained, for the same reasons in terms of phase shift. The 2.15 Hz and 2.65 Hz contributions are very well captured in the BS approach both in terms of magnitude and phase shift.

4.3. Comparison of linear and band superposition approach in the simulation of bridge response to turbulent wind As a final step of the present paper, numerical simulations of the deck response under the action of a turbulent wind are

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G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

Table 7 Structural properties. Mode

Natural frequency (Hz)

Modal mass

Damping coefficient

Mode shape y-component

Mode shape z-component

Mode shape y-component

1 2 3

0.0927 0.1977 0.2418

6.7e3 kg/m 6.7e3 kg/m 4.7e6 kg m2/m

0.01 0.01 0.01

1 0 0

0 1 0

0 0 1

performed using the linear and the BS approach. Full-scale simulations are performed considering a suspended deck sectional model run over by a fully turbulent wind completely correlated along the deck length. The proposed simulation represents the basic approach for the simulation of a full bridge. For a real structure, in fact, the deck aerodynamics is considered two-dimensional and the complete wind-deck interaction is computed by dividing the whole deck length in several segments and summing up the contribution of each single segment. Each segment is subjected to aerodynamic forces arising from the wind buffeting terms and from the motion induced terms that are modeled taking into account the part of the wind time–space field that is interacting with the considered deck segment. In this way it is possible to consider the spatial correlation of the wind velocity components that is neglected in the present simulation where the contribution of a single segment is computed. Nevertheless the proposed simulation, compared to the previous ones, allows to introduce the following elements of complexity in the deck interaction with a turbulent wind: 1. The turbulent wind is characterized by a full spectrum both of the horizontal and vertical velocity components and not by single harmonics on the vertical component only; 2. Buffeting terms and motion induced terms are contemporary exciting the deck. Simulations are performed under the following assumptions:

 The structural properties of the deck are those reported



in Table 7, corresponding to typical figures of full scale suspension bridges. The structure dynamics is modeled considering the first three modes (horizontal, vertical and torsional) that are rigid mode shapes for the sectional model. The turbulence characteristics of the incoming wind are reported in Table 8 and are representative of a real full scale situation measured on the Messina Strait site. A time history of the wind velocity components is numerically generated according to a wave superposition approach (Deodatis, 1996; Ding et al., 2006) imposing a Von Karman power spectral density and a random phase. The power spectral density (PSD) function of the vertical component of the wind velocity is reported in Fig. 20, in non-dimensional form overlapped to vertical dashed lines representing the three natural frequencies of the structure. A vertical red solid line indicates the cut-off frequency used to separate the HF band from the LF band in the BS approach. The cutn off frequency is chosen at V n ¼ 10 ðf ¼ 0:1Þ. According to the band separation, the wind time history is filtered to separate the LF and the HF contribution. The LF contribution is reported in Fig. 21 in terms of variation of equivalent angle of attack produced by the vertical velocity component (w/V). From the only contribution of the wind velocity components, neglecting the deck motion terms that are assumed to be small for an aerodynamic stable project, it is visible that the fluctuation of the angle

Table 8 Turbulent wind characteristics. Velocity component

IT

x

u w

13.80% 6.90%

177 m 17.7 m



Lu,w

of attack may reach up to 61 asking for a nonlinear approach in modeling the aerodynamic force terms. Simulations are performed considering three different mean wind speeds (30, 60, and 80 m/s) in order to investigate the deck response modeling the deck aerodynamic forces at different reduced velocities. For all the three mean wind speed the turbulence characteristics are kept constant.

Table 9 synthesizes the numerical results in terms of standard deviation and extreme values of deck motion. As an example, for additional comparison of linear and nonlinear results, Fig. 22 shows the PSD of the deck response for the case V¼60 m/s, in terms of deck rotation and vertical displacement. It is possible to observe how the adoption of the BS nonlinear approach may produce larger or smaller results, if compared to the linear model approach, depending on the considered mean wind speed. It is difficult to provide a univocal interpretation of the results because of the complexity introduced by all the nonlinear effects. The two most important effects investigated through the deterministic approach previously described are the production of sub- and super-harmonics and the introduction of the aerodynamic coefficients dependency from the instantaneous angle of attack. The two aspects are contemporary present and are produced by each single harmonic contribution composing the wind spectrum. Separating the buffeting terms from the motion induced one, it is expected that the differences between the linear approach and the BS approach are only related to the dependency of the admittance function from the angle of attack that is fixed and therefore it does not depend on the mean wind speed apart from the 12rV 2 term. The motion induced terms depend on how the structural modes contribute to the deck motion, and, considering different mean wind speeds they are therefore related to the different reduced velocities at which they are operating. Regardless the sub- and super-harmonics that will be summed up to the direct buffeting terms, the aerodynamic coefficient dependence on the angle of attack may play a different rule depending on the reduced velocity. For reduced velocity where the dependency is small the differences between the linear and the nonlinear approaches tends to be limited. When the dependence on the angle of attack is large the nonlinear approach may produce both larger and smaller results than the linear ones depending if the value of the aerodynamic coefficients for angles of attack different from the

G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

57

Fig. 20. Spectrum of the vertical component of the wind speed (vertical red solid line, cut-of frequency between LF and HF; vertical dashed black lines, first bridge natural frequencies). (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this article.)

Fig. 21. Instantaneous LF angle of attack.

Table 9 Summary of numerical simulations results. Wind speed (m/s)

30

Model

Lin

BS

Lin

BS

Lin

BS

sy (m) sz (m) sy (deg)

1.1 0.6 0.5 4.0 2.4 2.1

1.2 0.7 0.8 4.0 3.1 3.4

3.1 1.4 1.1 10.6 5.6 4.2

2.9 1.6 1.1 10.0 7.6 5.3

4.9 1.9 0.9 16.6 8.2 3.9

4.6 2.1 1.3 15.9 9.2 5.4

maxy (m) maxz (m) maxy (deg)

60

mean one are higher or smaller and on how they are distributed around the mean angle one.

5. Conclusions Experimental tests in wind tunnel demonstrated that low frequency fluctuations of the angle of attack influence higher frequency buffeting and motion induced forces, if unsteady coefficients have a dependence upon the mean angle of attack. In this

80

case, linear approaches to aerodynamic force modeling fail and a nonlinear approach is necessary to model the phenomenon. The proposed rheological model efficiently implements the band superposition hypothesis and allows to more correctly simulate the nonlinear dependence upon both reduced frequency and low frequency variation of the angle of attack. The numerical model has been validated using specific wind tunnel tests to reproduce BS conditions similar to the full scale one, where the larger fluctuations of the turbulent wind are responsible for large variations of the angle of attack.

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G. Diana et al. / J. Wind Eng. Ind. Aerodyn. 113 (2013) 40–58

incoming wind. The comparison with the results of a linear model shows that nonlinear effects can significantly influence the response of the bridge and they should be accounted for a more correct analysis of turbulence effects. References Argentini, T., Rocchi, D., Muggiasca, S., Zasso, A., 2012. Cross-sectional distributions versus integrated coefficients of flutter derivatives and aerodynamic admittances identified with surface pressure measurement. Journal of Wind Engineering and Industrial Aerodynamics 104–105, 152–158. Bocciolone, M., Cheli, F., Curami, A., Zasso, A., 1992. Wind measurements on the Humber bridge and numerical simulations. Journal of Wind Engineering and Industrial Aerodynamics 42, 1393–1404. Chen, X., Kareem, A., 2001. Nonlinear response analysis of long-span bridges under turbulent winds. Journal of Wind Engineering and Industrial Aerodynamics 89, 1335–1350. Chen, X., Kareem, A., 2003. Aeroelastic analysis of bridges: effects of turbulence and aerodynamic nonlinearities. Journal of Engineering Mechanics 129, 885–895. Chen, X., Matsumoto, M., Kareem, A., 2000. Time domain flutter and buffeting response analysis of bridges. Journal of Engineering Mechanics 126, 7–16. Deodatis, G., 1996. Simulation of ergodic multivariate stochastic processes. Journal of Engineering Mechanics 122, 778–787. Diana, G., Falco, M., Bruni, S., Cigada, A., Larose, G., Damsgaard, A., Collina, A., 1995. Comparisons between wind tunnel tests on a full aeroelastic model of the proposed bridge over stretto di messina and numerical results. Journal of Wind Engineering and Industrial Aerodynamics 54–55, 101–113. Diana, G., Resta, F., Rocchi, D., 2008. A new numerical approach to reproduce bridge aerodynamic non-linearities in time domain. Journal of Wind Engineering and Industrial Aerodynamics 96, 1871–1884. Diana, G., Resta, F., Zasso, A., Belloli, M., Rocchi, D., 2004. Forced motion and free motion aeroelastic tests on a new concept dynamometric section model of the messina suspension bridge. Journal of Wind Engineering and Industrial Aerodynamics 92, 441–462. Diana, G., Rocchi, D., Argentini, T., Muggiasca, S., 2010. Aerodynamic instability of a bridge deck section model: linear and nonlinear approach to force modeling. Journal of Wind Engineering and Industrial Aerodynamics 98, 363–374. Ding, Q., Zhu, L., Xiang, H., 2006. Simulation of stationary gaussian stochastic wind velocity field. Wind and Structures, An International Journal 9, 231–243. Wu, T., Kareem, A., 2011. Modeling hysteretic nonlinear behavior of bridge aerodynamics via cellular automata nested neural network. Journal of Wind Engineering and Industrial Aerodynamics 99, 378–388. Zasso, A., 1996. Flutter derivatives: advantages of a new representation convention. Journal of Wind Engineering and Industrial Aerodynamics 60, 35–47.

Fig. 22. Comparison of the Power Spectral Densities of the deck response using linear and nonlinear approach: deck rotation y (a) and vertical displacement (b).

Full scale low frequency fluctuations of the angle of attack are principally generated by large scale turbulent components of natural wind. The proposed model is therefore used to simulate the dynamic response of a full scale bridge to an highly turbulent