Identification of aerodynamic admittance functions of a flat closed-box deck in different grid-generated turbulent wind fields

Abstract

Wind tunnel tests of synchronous measurements of the buffeting lifts and moments on six separated strips of a motionless 1:45 scale sectional model of a flat closed-box bridge deck were carried out in three turbulent flow conditions. The longitudinal and vertical components of turbulent wind speed were also measured synchronously with the buffeting forces at the positions just over the windward edges of the six strips. Three typical methods, including the auto-spectral method based on the measured auto-spectrum of buffeting force (auto-spectrum method), the cross-spectral method based on the measured cross-spectra between buffeting force and fluctuating turbulence components (cross-spectrum method), and the colligated least square method based on a colligated residue of the above-mentioned auto- and cross-spectra (colligated least square method), were then used, respectively, to identify the aerodynamic admittance functions of the flat closed-box deck. The aerodynamic admittance function components of a buffeting force with respect to the longitudinal and vertical components of fluctuating wind speed were effectively separated using the colligated least square method while only a weighted average value, often called the equivalent aerodynamic admittance function, was obtained using the auto-spectrum method. Meanwhile, the reproduced auto-spectra of the buffeting forces had very high accuracies when the aerodynamic admittance functions identified with the colligated least square method were employed, but deviated significantly from the measured ones when the aerodynamic admittance functions identified with the cross-spectrum method were used. Furthermore, the influence of the span-wise position of the wind probe on the values of aerodynamic admittance function module was confirmed to be very small. Finally, the aerodynamic admittance function components with respect to the longitudinal and vertical fluctuating wind speeds were found to be significantly affected by both the intensities and integral scales of the oncoming turbulence wind.

Keywords

aerodynamic admittance function buffeting force buffeting force coefficient flat closed-box bridge deck grid-generated turbulent wind field integral scale of turbulence turbulent intensity

Introduction

In buffeting analyses of long-span bridges, aerodynamic admittance functions (AAFs) are often adopted to consider the effects of the spatial variation and incomplete correlation of the turbulent wind speeds surrounding the cross section of bridge deck and to correct approximately the unsteadiness on the quasi-steady model of buffeting forces. On this account, the coefficient spectra of the wind-induced buffeting force on a bridge deck have conventionally been expressed with equations (1) to (4) (Chen et al., 2001; Tubino, 2005; Zhu, 2002)

\begin{matrix} S_{C_{F_{b}}} (K) = {{a_{F}}^{2} {χ_{Fu}}^{2} S_{u} (K) + {b_{F}}^{2} {χ_{Fw}}^{2} S_{w} (K) \\ + 2 a_{F} b_{F} [χ_{Fu}^{*} χ_{Fw} S_{uw} (K) + χ_{Fw}^{*} χ_{Fu} S_{wu} (K)]} \end{matrix}

(1)

C_{F_{b}} = {\begin{matrix} F_{b} / (0.5 ρ U^{2} B), F = D, L \\ F_{b} / (0.5 ρ U^{2} B^{2}), F = M \end{matrix}

(2)

a_{F} = 2 C_{F} (θ_{0}) / U

(3)

b_{F} = {\begin{matrix} [C_{D}' (θ_{0}) - C_{L} (θ_{0})] / U, F = D \\ [C_{L}' (θ_{0}) + C_{D} (θ_{0})] / U, F = L \\ [C_{M}' (θ_{0})] / U, F = M \end{matrix}

(4)

where $F = D, L, M$ ; $F_{b}$ is the buffeting force on the bridge deck per unit span length and can be buffeting drag ( $D_{b}$ ), lift ( $L_{b}$ ), or torsional moment ( $M_{b}$ ), respectively; $C_{F_{b}}$ is the buffeting force coefficient; $C_{F}$ and $C'_{F}$ are the aerodynamic force coefficient and its derivative with respect to the wind attack angle, respectively; $θ_{0}$ is the initial attack angle of mean wind; $K = fB / U$ is the reduced frequency linked to the bridge deck width B; $ρ$ is the air density; U is the mean wind speed; f is the frequency (Hz); $S_{u}$ and $S_{w}$ are the power spectral densities of the longitudinal and vertical velocity components of turbulence u and w, respectively; $S_{uw}$ and $S_{wu}$ are the cross-spectra between u and w, respectively, and $S_{uw} = S_{wu}^{*}$ ; $χ_{Fu}$ and $χ_{Fw}$ are the AAFs of buffeting forces due to u and w, respectively; the superscript $*$ represents the conjugate operation of AAFs or cross-spectra.

In some literatures (Davenport, 1962; Davenport et al., 1992; Larose, 1997; Scanlan, 1993, 2000), approximate expressions about $b_{F}$ for drag and lift $D, L$ , which is slightly different from equation (4), can be found.

Actually, AAFs measure the ratios of the aerodynamic forces in fluctuating flows to their quasi-static values which are commonly obtained through wind tunnel test in smooth flow for a bridge deck (Davenport, 1962). Liepmann approximation of Sears function is the most common form of the lift and moment AAFs for thin airfoils or flat plates in fully correlated vertical gusts with sinusoidal fluctuations (Liepmann, 1952). Because of the bluff body feature of bridge decks and the complicacy of turbulence wind in the atmospheric boundary, AAFs of bridge deck cannot be simply expressed by the Sears function, but must be measured through wind tunnel test or computational fluid dynamics (CFD) simulation in conjunction with a proper identification algorithm. Up to now, there have been several algorithms for the AAF identification based on tested fluctuating aerodynamic forces and wind velocities, such as the auto-spectrum method (ASM) based on the measured auto-spectra of buffeting forces which is also called the equivalent AAF method (Gu and Qin, 2004), the zero-separation method (ZSM; Chen et al., 2009), the separated frequency-by-frequency method (SFFM) based on tests in vibrating grid–generated turbulent flows (Han et al., 2010), the cross-spectrum method (CSM) based on the measured cross-spectra between buffeting force and fluctuating wind speed components (Zhao and Ge, 2015), and the colligated least square method (CLSM) based on a colligated residue of the above-mentioned auto- and cross-spectra (Xu et al., 2014; Zhu et al., 2016). To identify AAFs, it is necessary to measure simultaneously buffeting forces acting on a sectional model and fluctuating velocity components of oncoming turbulent wind in a wind tunnel.

Buffeting forces can be obtained in wind tunnel tests through various ways, such as direct measurement of fluctuating forces with dynamic force balances, fluctuating pressure measurement in conjunction with pressure-integral technique, and systematic identification based on random displacement or acceleration responses of the sectional model to the turbulent wind. The third way is an indirect method compared with the first two ways (Gu and Qin, 2004), and the stability of the results obtained through this way is deemed to be uncertain (Zhao and Ge, 2015). Since the aerodynamic forces can be obtained by integrating the distributed pressures on a cross-sectional strip, the AAFs could also be identified through a pressure measurement test (Davenport et al., 1992; Haan, 2000; Larose, 1997; Ma et al., 2013). However, the pressure measurement approach has a few disadvantages compared with the force measurement approach, especially for bridge decks with wind barriers, crash barriers, and other ancillaries, on which the pressure measurement can be hardly conducted.

In traditional wind tunnel tests for AAF identification, the total fluctuating buffeting forces acting on the entire sectional model or a fairly long segment of the model are usually measured with a proper dynamometric system (Cigada et al., 2001; Diana et al., 2002, 2004, 2015; Hatanaka and Tanaka, 2002; Jancauskas and Melbourne, 1986; Matsuda et al., 1999; Sankaran and Jancauskas, 1992; Sarkar, 1992; Zhao and Ge, 2015). Then the distributed buffeting forces acting on cross-sectional strips of the model, needed in calculating AAFs, are often regarded to be equal to the measured total buffeting forces divided by the length of the model or the measured segment. This obviously implies an assumption that the distributed buffeting forces on different strips are fully correlated within the whole model or measured segment. However, this assumption is not strictly true because the fluctuating wind speeds as well as the fluctuating buffeting forces are stochastic, thus are actually partially correlated along the model span. In this connection, Xu et al. (2014) and Zhu et al. (2016) derived relevant formulae to consider the behavior of the span-wise partially correlation of the distributed buffeting forces when calculating the auto-spectra of the distributed buffeting forces and the cross-spectra between the distributed buffeting forces and the fluctuating wind speeds of the oncoming turbulence. Nevertheless, an additional synchronous measurement of pressures on multi-strips of another identical model is needed to obtain approximately the span-wise correlation information of the distributed buffeting forces. To avoid the additional pressure measurement test, a new technique of synchronous measurements of buffeting forces on narrow strips using a kind of miniature three-component dynamic force balances (Gao and Zhu, 2016) was elaborately developed by Zhu and his cooperators (Yan et al., 2017). The span-wise length of each deck model strip was only 30 mm and was notably smaller than the span-wise integral length scale of turbulence; therefore, the distributed buffeting forces could be considered to be fully correlated within such a small length. Thus, the AAFs could be directly identified from the measured buffeting force on the measured narrow strips. The purpose of carrying out the multi-strip force measurement in that test was to investigate the span-wise correlation of buffeting forces and will not be discussed in this article.

Jancauskas and Melbourne (1986) and Sankaran and Jancauskas (1992) measured the AAFs for a range of rectangular cylinders in smooth flow and grid-generated turbulent flows with different turbulence intensities, pointing out that the AAFs of rectangular sections rely on the chord-to-thickness ratio of the section and the turbulence intensity. Some other research results also show that the turbulent intensity of grid-generated turbulent wind field exerts a certain influence on the identified AAFs (Wen, 2008; Zhao and Ge, 2015). Larose (1997) directly measured the AAFs for typical closed-box girder bridge decks similar to the Great Belt East Bridge. Four different width-to-depth ratios and three turbulent flow fields with similar turbulence intensities but three different integral scales of turbulence were considered in his research. Hui (2006) studied the AAFs of typical twin-deck box girder sections similar to the Stonecutters Bridge. Five different gap widths and two grid-generated turbulent wind fields with similar turbulence intensities but different integral scales of turbulence were taken into account in his tests to investigate the influence of gap width and integral scale of turbulence on AAFs. The research results presented by both Larose and Hui demonstrated that the AAFs depend on the shape of bridge deck as well as the integral scale of turbulence. As a summary, the AAFs mainly depend on the aerodynamic shape of bridge deck, while they also influenced by the turbulent intensity and the integral length scale of turbulence (Li et al., 2016; Tomasini and Cheli, 2013).

Nowadays, passive grid-generated turbulent wind field is still the prevailing one in wind tunnel tests for AAF identification. However, as well known, it has evident disadvantages that its integral length scale of turbulence is often small and is difficult to be adjusted. Hence, research results about the influence rules of the integral scale and intensity of turbulence on AAFs are still insufficient and need to supplement further. This work will be very useful for improving the prediction accuracy of buffeting responses of long-span bridges and is a major task of this study. However, a proper method for identifying AAFs should be selected heretofore from the currently prevailing methods by comparison analyses.

Brief description of identification approaches

The ASM for AAF identification is only based on the measured auto-spectrum of buffeting force and needs an assumption that the admittance of a buffeting force due to the longitudinal fluctuating velocity (u) is as the same as that due to the vertical fluctuating velocity (w), that is

χ_{Fu} = χ_{Fw} = χ_{F}

(5)

where $χ_{F}$ is often called the equivalent aerodynamic admittance function (EAAF) and is actually a weighted average value of $χ_{Fu}$ and $χ_{Fw}$ .

Thus, $χ_{F}$ can be determined approximately by the following equation (Gu and Qin, 2004; Haan, 2000; Larose, 1997)

{| χ_{F} |}^{2} = \frac{S_{C_{F}}}{{a_{F}}^{2} S_{u} + {b_{F}}^{2} S_{w}}

(6)

where the influence of the cross-spectrum between u and w on the auto-spectrum of buffeting forces is neglected, and the error of this neglect is about 5%–7% as reported by Kumarasena (1989).

To distinguish $χ_{Fu}$ and $χ_{Fw}$ , a CSM was then adopted by some researchers (Jin, 2003; Ma et al., 2013; Zhao and Ge, 2015) based on the measured cross-spectra between the fluctuating force coefficient $C_{F}$ and each of u and w. The cross-spectral equations for solving AAFs are as follows

{\begin{matrix} S_{C_{F} u} = a_{F} χ_{Fu}^{*} S_{u} + b_{F} χ_{Fw}^{*} S_{wu} \\ S_{C_{F} w} = a_{F} χ_{Fu}^{*} S_{uw} + b_{F} χ_{Fw}^{*} S_{w} \end{matrix}

(7)

The two AAF components can then be calculated with the following expressions

{\begin{matrix} χ_{Fu}^{*} = \frac{S_{w} S_{C_{F} u} - S_{wu} S_{C_{F} w}}{a_{F} (S_{u} S_{w} - S_{wu} S_{uw})} \\ χ_{Fw}^{*} = \frac{S_{u} S_{C_{F} w} - S_{uw} S_{C_{F} u}}{b_{F} (S_{u} S_{w} - S_{wu} S_{uw})} \end{matrix}

(8)

where $S_{C_{F} a} (a = u, w)$ are the cross-spectral densities of the buffeting force coefficient $C_{F}$ and the fluctuating wind velocity component a.

However, the correlation between the buffeting force and the fluctuating wind is usually fairly weak and thus the identified AAFs show rather strong random behavior. Therefore, the auto-spectra of the fluctuating force reproduced using the identified AAFs often deviate far away from the measured ones (Hui, 2006; Xu et al., 2014). This will significantly reduce the analysis accuracy of buffeting responses of long-span bridges.

To distinguish the AAF components with respect to u and w, that is, $χ_{Fu}$ and $χ_{Fw}$ , and at the same time to ensure the accuracy of the reconstructed auto-spectra of the fluctuating forces using the identified AAFs, a new CLSM for identifying AAFs of bridge decks was developed by Zhu and his students (Li, 2009; Xu et al., 2014; Zhu et al., 2016). The AAF components are identified by finding the minimum value of a colligated residue comprising the auto-spectral residue of buffeting force coefficient and the cross-spectral residues between the buffeting force coefficient and every of u and w. The colligated residue function in CLSM is defined as follows

\begin{matrix} ε_{F} (χ_{Fu}^{Re}, χ_{Fu}^{Im}, χ_{Fw}^{Re}, χ_{Fw}^{Im}) = (w_{1} ε_{F 1})^{2} + w_{2} ({ε_{F 2}}^{2} + {ε_{F 3}}^{2}) \\ + w_{3} ({ε_{F 4}}^{2} + {ε_{F 5}}^{2}) \end{matrix}

(9)

\begin{matrix} ε_{F 1} = - S_{C_{F}} + {a_{F}}^{2} {χ_{Fu}}^{2} S_{u} + {b_{F}}^{2} {χ_{Fw}}^{2} S_{w} \\ + 4 a_{F} b_{F} [(χ_{Fu}^{Re} χ_{Fw}^{Re} + χ_{Fu}^{Im} χ_{Fw}^{Im}) S_{uw}^{Re} + (χ_{Fu}^{Im} χ_{Fw}^{Re} - χ_{Fu}^{Re} χ_{Fw}^{Im}) S_{uw}^{Im}] \end{matrix}

(10)

\begin{matrix} ε_{F 2} = - S_{C_{F} u}^{Re} \\ + [a_{F} (χ_{Fu}^{Re} S_{u}^{Re} - χ_{Fu}^{Im} S_{u}^{Im}) + b_{F} (χ_{Fw}^{Re} S_{wu}^{Re} + χ_{Fw}^{Im} S_{wu}^{Im})] \end{matrix}

(11)

\begin{matrix} ε_{F 3} = - S_{C_{F} u}^{Im} \\ + [- a_{F} (χ_{Fu}^{Re} S_{u}^{Im} + χ_{Fu}^{Im} S_{u}^{Re}) + b_{F} (χ_{Fw}^{Re} S_{wu}^{Im} - χ_{Fw}^{Im} S_{wu}^{Re})] \end{matrix}

(12)

\begin{matrix} ε_{F 4} = - S_{C_{F} w}^{Re} \\ + [a_{F} (χ_{Fu}^{Re} S_{uw}^{Re} - χ_{Fu}^{Im} S_{uw}^{Im}) + b_{F} (χ_{Fw}^{Re} S_{w}^{Re} + χ_{Fw}^{Im} S_{w}^{Im})] \end{matrix}

(13)

\begin{matrix} ε_{F 5} = - S_{C_{F} w}^{Im} \\ + [- a_{F} (χ_{Fu}^{Re} S_{uw}^{Im} + χ_{Fu}^{Im} S_{uw}^{Re}) + b_{F} (χ_{Fw}^{Re} S_{w}^{Im} - χ_{Fw}^{Im} S_{w}^{Re})] \end{matrix}

(14)

where $w_{i} (i = 1, 2, 3)$ are the weighting factors; the superscripts “Re” and “Im” represent the real and imaginary parts of a complex quantity or variable, respectively. The aerodynamic admittance $χ_{Fa}$ of $F (= D, L, M)$ associated with $a (= u, w)$ is complex in general

χ_{Fa} (K) = χ_{Fa}^{Re} (K) + j χ_{Fa}^{Im} (K)

(15)

where $j = \sqrt{- 1}$ .

Equation (15) is often rewritten with the following form

χ_{Fa} (K) = | χ_{Fa} (K) | \cdot Φ_{Fa} (K)

(16)

where $| χ_{Fa} (K) |$ and $Φ_{Fa} (K)$ are the module function and the phase function of $χ_{Fa}$ , respectively, and defined as

| χ_{Fa} (K) | = \sqrt{χ_{Fa}^{Re} {(K)}^{2} + χ_{Fa}^{Im} {(K)}^{2}}

(17)

Φ_{Fa} (K) = \tan^{- 1} [\frac{χ_{Fa}^{Im} (K)}{χ_{Fa}^{Re} (K)}]

(18)

Experimental setup

The cross section of a flat closed-box deck is shown in Figure 1. All the wind tunnel tests were carried out in the TJ-2 Wind Tunnel, and they were typically conducted at a mean wind speed of 12 m/s. The sectional model was constructed at a length scale of 1:45. Its whole length (L), width (B), and depth (D) were 1740, 775.6, and 77.8 mm, respectively; thus, the width-over-depth ratio (B/D) was 9.71. The schematic layout of the experimental model in the wind tunnel is shown in Figure 2. Further details about the modeling, the instrumentation, the grid setup, the force balance, and the measurement strip arrangement are available in Yan et al. (2017).

Figure 1.

Cross section of the flat closed-box deck (mm).

Figure 2.

Schematic diagram of the experimental setup (model, force balances, grids, wind speed probes).

Three types of grid-generated turbulent wind fields were employed in the tests. They were achieved by adjusting the strip width of the grids and the distance between the grid section and the front wind fairing of the deck. As shown in Table 1, the turbulent wind fields generated by Grid B and Grid C possessed similar integral length scales of turbulence but different turbulent intensities, while those generated by Grid B and Grid D had similar turbulent intensities but different integral length scales. An isotropic turbulence model based on the von Karman spectrum (Mann, 1994) is used here for fitting the measured spectra of u, v, and w components of turbulent wind. A detailed discussion of the applicability of isotropic turbulence model in wind engineering can be found in the literatures (Larose, 1997; Mann, 1994).

Table 1.

Parameters of turbulence over Section D in three kinds of wind fields generated by Grids B–D.

Inflow	d (m)	v ₁ (m)	v ₂ (m)	h (m)	U (m/s)	I_u (%)	$L_{u}^{x}$ (m)	$L_{\max, w}^{x}$ (m)	$I_{v}$ (%)	$L_{v}^{x}$ (m)	$L_{\max, v}^{x}$ (m)	$I_{w}$ (%)	$L_{w}^{x}$ (m)	$L_{\max, w}^{x}$ (m)
Grid B	4.02	0.15	0.15	0.15	12.1	12.8	0.231	0.253	10.8	0.086	0.130	10.9	0.081	0.122
Grid C	6.62	0.10	0.10	0.10	12.2	5.8	0.237	0.260	5.3	0.091	0.136	5.0	0.083	0.125
Grid D	6.62	0.15	0.20	0.15	12.1	13.9	0.363	0.397	12.4	0.161	0.242	12.4	0.142	0.214

The spectrum for u component can be expressed as

S_{u} (k_{1}) = \frac{9}{55} α ε^{2 / 3} \frac{1}{{({L_{1}}^{- 2} + k_{1}^{2})}^{5 / 6}}

(19)

And that for both the v and w components is as follows

S_{a} (k_{1}) = \frac{3}{110} α ε^{2 / 3} \frac{3 {L_{1}}^{- 2} + 8 k_{1}^{2}}{{({L_{1}}^{- 2} + k_{1}^{2})}^{11 / 6}} a = v, w

(20)

Here, $k_{1}$ is a wave number ( $2 π f / U$ ); $ε$ represents the rate of viscous dissipation of turbulent kinetic energy; $α$ is an empirical coefficient; $L_{1}$ is a kind of the length scale. In practice, $α ε^{2 / 3}$ and $L_{1}$ can be fitted based on the measured data of spectra.

The following relationships exist between $L_{1}$ and the integral length scale $L_{a}^{x} (a = u, v, w)$ for the isotropic turbulence

{\begin{matrix} L_{u}^{x} = L_{1} \frac{1}{\sqrt{π}} \frac{Γ (1 / 3)}{Γ (5 / 6)} \approx 0.746 L_{1} \\ L_{a}^{x} = \frac{1}{2} L_{u}^{x} = 0.373 L_{1}, a = v, w \end{matrix}

(21)

$L_{\max, a}^{x} (a = u, v, w)$ is denoted as a characteristic length of eddy and $1 / L_{\max, a}^{x}$ represents the wave number at the maximum $k_{1} S_{a} (K_{1})$ (Mann, 1994). It is a more appropriate characteristic length to describe a turbulent flow field and will be used as a reference length later in this study. $L_{\max, a}^{x} (a = u, v, w)$ is related to $L_{1}$ for the modified von Karman spectrum by the following expressions

{\begin{matrix} L_{\max, u}^{x} = {(\frac{2}{3})}^{\frac{1}{2}} L_{1} \approx 0.816 L_{1} \\ L_{\max, a}^{x} = \frac{2}{{(6 + 3 \sqrt{5})}^{\frac{1}{2}}} L_{1} \approx 0.561 L_{1}, a = v, w \end{matrix}

(22)

Based on equations (19) and (20), the measured data of normalized auto-spectra of wind turbulence, $f S_{a} / σ_{a}^{2}$ , were then fitted using the MATLAB software and a non-linear least square fitting (NLSF) method to determine the parameters of $α ε^{2 / 3}$ and $L_{1}$ , where a = u, v, w, and $σ_{a}^{2}$ is the variance of the turbulence component a. The fitted results are shown in Figure 3 for the turbulence at Section D generated by Grid B. It can be seen that equations (19) and (20) fitted the measured data quite well both in the low- and high-frequency regions. The fitted parameters $L_{1}$ were 0.31, 0.23, and 0.22 m, respectively, for the fluctuating wind components of u, v, and w, where the value for u is largest. The fitted values of $α ε^{2 / 3}$ for the three velocity components were similar to each other. The peak frequency ( $f_{p}$ ) of the fitted curve of $f S_{a} / σ_{a}^{2}$ is corresponding to the eddy with most energy in turbulence. The values of $f_{p}$ were 7.6, 14.7, and 15.8 Hz, respectively, for the three wind components of u, v, and w, where the value for u is smallest. The normalized auto-spectrum of w was found similar to that of v. The integral length scale, $L_{a}^{x} (a = u, v, w)$ , and the characteristic length of eddy, $L_{\max, a}^{x} (a = u, v, w)$ , were then calculated by the isotropic turbulence model. The obtained values of the various parameters of turbulence over Section D are listed in Table 1 for the three kinds of wind field generated by Grids B–D.

Figure 3.

Normalized auto-spectra of wind turbulence over Section D in Grid B turbulence flow: (a) longitudinal component, (b) lateral component, and (c) vertical component.

Major test results

The tests focused on measuring the fluctuating lifts and torsional moments acting on the six strips of the motionless bridge deck model using six miniature five-component force balances. The tests were carried out in the three types of grid-generated turbulence wind fields with a mean wind speed of about 12.0 m/s. Both the fluctuating forces and wind speeds were acquired at a sampling frequency of 200 Hz, and the sample length was 131,072 data points corresponding to 655 s. The MATLAB software and Welch’s averaged modified periodogram method were adopted to estimate the power spectral densities of the measured fluctuating forces and wind speeds. The piecewise smoothing method was then applied to the spectral analysis of the measured data of both the fluctuating forces and wind speeds. Every acquired data sample was divided into 16 segments with an overlap length being a half of the segment. The block size for the fast Fourier transform (FFT) was equal to 8192. The Hamming window was also used in the spectral analyses of every data segments to reduce the leakage of signals in the frequency domain from one band to another.

Identified AAFs with different approaches

The measured fluctuating buffeting lift $F_{L}$ and moment M can be rendered dimensionless and expressed in terms of the coefficients of buffeting lift and moment ( $C_{L_{b}}$ and $C_{M_{b}}$ ) by equation (2). For long-span bridges under strong winds, the response components with reduced frequencies $K = fB / U \leq 1.0$ are often dominant (Ito et al., 2014). Therefore, the cut-off frequency for the AAFs in this study was set as 15.6 Hz, corresponding to a cut-off reduced frequency of 1.0.

The aerodynamic coefficients of the static lift and torsional moment of the bridge deck obtained by Zhu and Guo (2009) through force measurement test of sectional model were used here for the AAF identification. This is because the miniature five-component miniature dynamic force balances used in this study are of piezoelectric type and can measure only the dynamic component of forces and moments without the mean component. The static aerodynamic coefficients at the zero attack angle of wind were $C_{D} (0^{0}) = 0.0585$ , $C_{L} (0^{0}) = - 0.032$ , and $C_{M} (0^{0}) = 0.0138$ , and their derivatives were $C'_{L} (0^{0}) = 4.23$ and $C'_{M} (0^{0}) = 0.650$ .

Figure 4 shows the lift and moment AAFs with respect to the longitudinal component of turbulent wind speed u ( ${| χ_{Lu} |}^{2}$ and ${| χ_{Mu} |}^{2}$ ), which were identified through the tests in the three types of grid-generated turbulent wind fields using the foregoing three identification methods, respectively. Figure 5 displays those with respect to the vertical component of turbulent wind speed w ( ${| χ_{Lw} |}^{2}$ and ${| χ_{Mw} |}^{2}$ ). The square of AAF module is chosen to carry out comparisons among the identified AAFs with different identification methods in this section because ASM can only provide the module of the equivalent AAF.

Figure 4.

Aerodynamic admittances with respect to the longitudinal component of fluctuating wind velocity u: (a) lift AAFs in Grid B wind field, (b) moment AAFs in Grid B wind field, (c) lift AAFs in Grid C wind field, (d) moment AAFs in Grid C wind field, (e) lift AAFs in Grid D wind field, (f) moment AAFs in Grid D wind field.

Figure 5.

Aerodynamic admittances with respect to the vertical component of fluctuating wind velocity w: (a) lift AAFs in Grid B wind field, (b) moment AAFs in Grid B wind field, (c) lift AAFs in Grid C wind field, (d) moment AAFs in Grid C wind field, (e) lift AAFs in Grid D wind field, (f) moment AAFs in Grid D wind field.

It can be seen from Figure 4 that ${| χ_{Fu} |}^{2}$ (F = L, M) identified with ASM is significantly smaller than the corresponding ones identified with CSM and CLSM. This is because ASM assumes that $χ_{Fu} = χ_{Fw}$ (F = L, M). Namely, the identified equivalent AAF by ASM is actually a weighted mean value of $χ_{Fu}$ and $χ_{Fw}$ and is much closer to $χ_{Fw}$ because the value of the aerodynamic static coefficient $C_{F}$ is generally very small at a small attack angle of wind for both the lift and torsional moment, and the weighting factor of $χ_{Fu}$ is hence significantly smaller than that of $χ_{Fw}$ in general. Furthermore, one can also find that the value of ${| χ_{Fu} |}^{2}$ (F = L, M) identified with CLSM is generally larger than that identified with CSM to some extent.

From Figure 5, it can be found that the value of ${| χ_{Fw} |}^{2}$ (F = L, M) identified with CSM is much smaller than the corresponding ones identified with the other two methods, that is, ASM and CLSM. Furthermore, the result demonstrates that the equivalent AAFs identified with ASM are very close to ${| χ_{Fw} |}^{2}$ obtained with CLSM.

The ${| χ_{Fw} |}^{2}$ of Sears function for flat plates (Sears, 1941) is also plotted in Figure 5. It can be seen that the values of ${| χ_{Fw} |}^{2}$ identified with ASM and CLSM share the changing trend with the Sears function at the most frequency points except for the low region of the reduced frequency K less than 0.1–0.2. This is because the concerned closed-box deck is rather flat and its aerodynamic property should be relatively close to a flat plate. The discrepancy of the identified value from Sears function within the low-frequency region may be due to the insufficiency of the integral length scales of the simulated turbulent wind field.

Figure 6 shows comparisons among the coefficient spectra of buffeting lift and moment on Section D measured in the three types of grid-generated turbulent wind fields and those reproduced with the AAFs identified with CSM and CLSM. It can be seen that although CSM can separate the AAF components with respect to the fluctuating wind velocity components of u and w, the auto-spectra of buffeting lift and moment coefficient reconstructed based on CSM deviate from the measured ones remarkably. This occurrence may be caused by the fact that the auto-spectral equations of buffeting forces are not included in CSM, and the correlation between the buffeting forces and the fluctuating wind velocities is generally rather weak, which may result in significant randomness and low accuracy of the identified AAFs. The auto-spectra of buffeting lift and moment coefficients reproduced based on CLSM agree with the measured ones very well because not only the equations of the cross-spectra between buffeting forces and fluctuating wind velocities but also the equations of the auto-spectra of buffeting forces are considered in CLSM.

Figure 6.

Comparison among the reproduced and measured spectra of buffeting force coefficients: (a) lift coefficient spectra in Grid B wind field, (b) moment coefficient spectra in Grid B wind field, (c) lift coefficient spectra in Grid C wind field, (d) moment coefficient spectra in Grid C wind field, (e) lift coefficient spectra in Grid D wind field, (f) moment coefficient spectra in Grid D wind field.

It is then evident that the CLSM of auto- and cross-spectra is the most suitable one among the three compared methods for AAF identification. Therefore, CLSM is going to be adopted in the next step.

Influence of span-wise relative position between force and wind measurements

How to select a proper measurement position of fluctuating wind speed in wind tunnel tests of synchronous measurement of fluctuating force and wind speed for AAF identification is always a perplexing task. This is because a too close position of wind probe to the model will lead to the measured wind speed disturbed significantly by the model, while a too far position from the model may result in a notable difference between the measured wind speed and the ideal incident wind speed in the case without the model.

Considering the flows over the center of the deck model is disturbed generally more severely than the flow over the windward edge of the model and the longitudinal interval between these two positions is only 377.8 mm, the wind measurement position in this study was then set right over the windward edge of the model. To determine a reasonable vertical distance between the wind measurement position and the model surface, a pretest on turbulent wind was carried out over the windward edge of the model within a range of the vertical distance from 220 to 420 mm. The details of the pretest are not presented here for space limitation. It was found through the pretest that wind characteristics did not vary significantly within the range of the vertical distance from 320 to 420 mm. On this account, the distance of 320 mm (4.11 times of the deck model depth) was finally chosen for the wind measurement or cobra probe position over the windward edge of the model, and the turbulent wind at this position was regarded to be able to represent the oncoming one approximately.

Furthermore, there were six measurement strips on the sectional model as shown in Figure 2, which were originally for investigating the span-wise correlation of buffeting force. Each of them was supported by a force balance fixed on the internal metal frame of the sectional model. The span-wise intervals (along with the y-axis) between two neighboring measurement sections were 120, 180, 31, 180, and 120 mm, respectively. Correspondingly, six cobra probes were set at the positions right over the windward edges of the measurement strips at a distance of 320 mm to measure the incident turbulent wind speeds over the six strips. The measured fluctuating wind velocities by four of these six wind probes (Probes A–D) and the measured fluctuating forces on Strip D by Balance D in the turbulent wind field of Grid B are then used in this section to investigate the influence of the span-wise relative position between the force and wind speed measurements on the identified results of AAFs. The purpose of this work is just to check whether the turbulent wind needs to be measured only at one position in such a kind of force measurement test on multi-strips.

Figure 7 shows the modules and phases of the identified $χ_{Lu}$ and $χ_{Lw}$ based on the measured fluctuating wind speed and lift data, respectively, using the four combinations of Balance D and each of Probes A–D in turbulent wind field of Grid B. It can be seen from Figure 7 that the influence of the change in the span-wise relative position between the wind and force measurements on the modules of the identified $χ_{Lu}$ and $χ_{Lw}$ is not significant, especially for $χ_{Lw}$ , although the span-wise relative positions between Balance D and Probes A, B, C, and D were very different. One can also find that the dispersion of the identified values of $| χ_{Lu} |$ is obviously more evident than that of $| χ_{Lw} |$ . This is because the weighting factor of $χ_{Lu}$ in the buffeting force auto-spectra as well as in the cross-spectra between the buffeting forces and the fluctuating wind speeds is significantly smaller than that of $χ_{Lw}$ , leading to that the target function of the colligated residue of CLSM is not sensitive to $χ_{Lu}$ .

Figure 7.

Modules and phases of lift AAFs identified based on the measured data using four combinations of Balance D with each of probes A–D in Grid B wind field: (a) module of AAF with respect to u, (b) module of AAF with respect to w, (c) phase of AAF with respect to u, (d) phase of AAF with respect to w.

From Figure 7, one can also find that both the phase functions of the identified $χ_{Lu}$ and $χ_{Lw}$ fluctuate dramatically with the change in the reduced frequency for all the four different combinations of Balance D and Probes A–D. Moreover, the change in the span-wise relative position between the wind and force measurements does influence the phase functions of the identified $χ_{Lu}$ and $χ_{Lw}$ . This kind of influence becomes more significant with the increase in the span-wise interval between the wind probe and the force measurements.

Figure 8 displays the modules and phases of the identified $χ_{Mu}$ and $χ_{Mw}$ based on the measured fluctuating wind speed and torsional moment data, respectively, using the four pairs of force balance and wind probe in turbulent wind field of Grid B. Similar results and conclusions to those for the lift AAFs can also be obtained for the moment AAFs.

Figure 8.

Modules and phases of moment AAFs identified based on the measured data using four combinations of Balance D with each of probes A–D in Grid B wind field: (a) module of AAF with respect to u, (b) module of AAF with respect to w, (c) phase of AAF with respect to u, (d) phase of AAF with respect to w.

Influences of turbulence characteristic parameters

By inspecting the AAFs’ curves, either obtained from analytical calculations or direct measurements, Larose (1997) proposed an empirical relationship between AAFs and turbulence parameters expressed as equation (23) for the lift and moment AAFs

{| χ_{F} (K, \frac{L_{\max, w}^{x}}{B}) |}^{2} = \frac{a (1 + \sqrt{K})}{1 + b K^{2}} (F = L, M)

(23)

where $L_{\max, w}^{x}$ represents the length of the eddy of turbulence, described in section “Experimental setup;”a and b are the parameters to be fitted.

Zhou (2011) proposed another more generalized empirical model expressed as equation (24) for the AAFs

{| χ_{F} (K) |}^{2} = \exp (a + \frac{b + cK}{1 + dK + e K^{2}})

(24)

where $F = L, M$ ; a, b, c, d, and e are five model parameters that need to be fitted using the experimental data.

The parameters in the two empirical models were fitted using the MATLAB software through a direct NLSF to the AAF data identified using CLSM. The target residue functions in NLSF were defined as follows:

For Larose’s model

R (a, b) = \sum_{i = 1}^{n} {[\frac{a (1 + \sqrt{K_{i}})}{1 + {bK}_{i}^{2}} - {| {\hat{χ}}_{F} (K_{i}) |}^{2}]}^{2}

(25)

For Zhou’s model

\begin{matrix} R (a, b, c, d, e) \\ = \sum_{i = 1}^{n} {[\exp (a + \frac{b + c K_{i}}{1 + d K_{i} + {eK}_{i}^{2}}) - {| {\hat{χ}}_{F} (K_{i}) |}^{2}]}^{2} \end{matrix}

(26)

where ${| {\hat{χ}}_{F} (K_{i}) |}^{2}$ is the square value of AAF module at the reduced frequency $K_{i}$ identified through the test; n is the total number of the data of the identified ${| {\hat{χ}}_{F} (K_{i}) |}^{2}$ . By finding the minimum values of the above target residue functions with the function of “fmincon” in MATLAB based on a “trust region reflective algorithm,” one can obtain the parameters a and b in Larose’s model or $a, b, c, d$ , and e in Zhou’s model.

Figure 9 displays the fitted results of the identified ${| χ_{Lu} |}^{2}$ and ${| χ_{Mu} |}^{2}$ through wind tunnel tests in the three types of grid-generated turbulent wind fields. The empirical model suggested by Zhou (2011) fits the measured AAFs well within whole concerned range of the reduced frequency, whereas the expression recommended by Larose (1997) fits the measured AAFs reasonably only in the range of relatively high frequencies, especially for ${| χ_{Lu} |}^{2}$ . The Larose model of AAFs is not able to describe the “fall-down” phenomenon of AAFs within the range of low reduced frequencies approaching zero.

Figure 9.

Lift and moment aerodynamic admittance components with respect to u in three grid-generated turbulences: (a) lift AAFs in Grid B wind field, (b) moment AAFs in Grid B wind field, (c) lift AAFs in Grid C wind field, (d) moment AAFs in Grid C wind field, (e) lift AAFs in Grid D wind field, (f) moment AAFs in Grid D wind field.

Figure 10 shows the fitted results of the identified ${| χ_{Lw} |}^{2}$ and ${| χ_{Mw} |}^{2}$ through wind tunnel tests in the three types of grid-generated turbulent wind fields. Once more, it can be seen that Zhou’s model fits the measured AAFs very well for all cases. Larose’s model also fits the measured AAFs quite well when the “fall-down” phenomenon of the measured AAFs within the low-frequency range is not significant; otherwise, the fitted AAFs with Larose’s model deviate clearly from the measured ones within the low-frequency range.

Figure 10.

Lift and moment aerodynamic admittance components with respect to w in three grid-generated turbulences: (a) lift AAFs in Grid B wind field, (b) moment AAFs in Grid B wind field, (c) lift AAFs in Grid C wind field, (d) moment AAFs in Grid C wind field, (e) lift AAFs in Grid D wind field, (f) moment AAFs in Grid D wind field.

As a result, the fitted AAFs with Zhou’s model are chosen in the next study. Figure 11 gives a comparison among the fitted AAFs with Zhou’s model with respect to the fluctuating wind velocities of u and w over Section D, obtained, respectively, in the three types of the simulated turbulent wind field, where all the other test parameters were kept unchanged. The shapes of ${| χ_{Lu} |}^{2}$ and ${| χ_{Mu} |}^{2}$ in Grid B and Grid C wind fields were quite similar within the range of higher reduced frequency $K > 0.2 - 0.3$ , and the values of ${| χ_{Lu} |}^{2}$ and ${| χ_{Mu} |}^{2}$ in Grid B wind field were larger than those in Grid C wind field. This occurrence is very likely associated with the fact that Grid B and Grid C had the similar integral length scales of turbulence but different turbulent intensities. The values of ${| χ_{Lu} |}^{2}$ in Grid B wind field were very close to that in Grid C wind field in the low reduced frequency range of $K < 0.2 - 0.3$ , whereas the gap of ${| χ_{Mu} |}^{2}$ in Grid B and Grid C wind fields was large, especially at very low reduced frequencies. It can also be found from Figure 11 that the magnitude of ${| χ_{Lu} |}^{2}$ and ${| χ_{Mu} |}^{2}$ in Grid D wind field is much larger than that in Grid B wind field within the whole frequency range concerned, but the gap between them slowly becomes smaller and smaller with the increase in the reduced frequency. Noticing that Grid D wind field possessed larger integral length scales than Grid B wind field, although they had similar turbulent intensities, it can then be inferred that the increase in integral length scales of oncoming flow will efficiently enhance the fluctuating intensity of the buffeting lift and torsional moment on the bridge deck. Furthermore, one can find that the variation patterns of ${| χ_{Lw} |}^{2}$ and ${| χ_{Mw} |}^{2}$ with the reduced frequency are similar to those for ${| χ_{Lu} |}^{2}$ and ${| χ_{Mu} |}^{2}$ , and the magnitudes of ${| χ_{Lw} |}^{2}$ and ${| χ_{Mw} |}^{2}$ are much smaller than those for ${| χ_{Lu} |}^{2}$ and ${| χ_{Mu} |}^{2}$ . The magnitudes of ${| χ_{Lw} |}^{2}$ and ${| χ_{Mw} |}^{2}$ in Grid D wind field are larger than those in Grid B wind field at the low reduced frequencies $K < 0.2 - 0.3$ , and smaller than those in Grid B wind field at higher reduced frequencies $K > 0.2 - 0.3$ . Furthermore, the patterns of ${| χ_{Lw} |}^{2}$ and ${| χ_{Mw} |}^{2}$ are obviously different from the patterns of ${| χ_{Lu} |}^{2}$ and ${| χ_{Mu} |}^{2}$ .

Figure 11.

Fitted AAFs with respect to the fluctuating velocities of u and w over Section D in the three types of turbulent wind fields: (a) fitted lift AAF with respect to u, (b) fitted moment AAF with respect to u, (c) fitted lift AAF with respect to w, (d) fitted moment AAF with respect to w.

As a summary, the lift and moment AAFs are not only dependent on the turbulent intensities of the oncoming flow but also affected by its integral length scales of turbulence.

Conclusion

The CLSM based on a colligated residue function of buffeting force auto-spectrum and cross-spectra between buffeting forces and turbulent wind speeds has been proved to be able to separate effectively the aerodynamic admittance components with respect to the longitudinal and vertical components of turbulent wind speed. It can also ensure the accuracy of the auto-spectra of buffeting forces at the same time.

The influence of span-wise relative position between the force and wind measurements on the module of AAFs has been confirmed to be very small.

The AAFs were found to significantly depend on the turbulent intensity and the integral length scale of turbulence of the incident wind.

Footnotes

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: the National Natural Science Foundation of China (grant nos 51478360, 51323013, and 91215302). Any opinions and concluding remarks presented in this article are entirely those of the writers.

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