Mean wind load induced incompatibility in nonlinear aeroelastic simulations of bridge spans

Abstract

Mean wind response induced incompatibility and nonlinearity in bridge aerodynamics is discussed, where the mean wind and aeroelastic loads are applied simultaneously in time domain. A kind of incompatibility is found during the simultaneous simulation of the mean wind and aeroelastic loads, which leads to incorrect mean wind structural responses. It is found that the mathematic expectations (or limiting characteristics) of the aeroelastic models are fundamental to this kind of incompatibility. In this paper, two aeroelastic models are presented and discussed, one of indicial-function-denoted (IF-denoted) and another of rational-function-denoted (RF-denoted). It is shown that, in cases of low wind speeds, the IF-denoted model reflects correctly the mean wind load properties, and results in correct mean structural responses; in contrast, the RF-denoted model leads to incorrect mean responses due to its nonphysical mean properties. At very high wind speeds, however, even the IF-denoted model can lead to significant deviation from the correct response due to steady aerodynamic nonlinearity. To solve the incompatibility at high wind speeds, a methodology of subtraction of pseudo-steady effects from the aeroelastic model is put forward in this work. Finally, with the method presented, aeroelastic nonlinearity resulted from the mean wind response is investigated at both moderate and high wind speeds.

Keywords

bridge / aerodynamics / nonlinear / aeroelastic model / Pseudo-steady / mean wind loads

Introduction

Exact assessment of the wind-resistant behaviors of long-span bridge decks involves generally description of several basic types of aerodynamic loads. They are mean wind loads, motion-induced (or self-excited) loads, buffeting loads, and vortex-induced loads. The mean wind loads, denoted in terms of steady aerodynamic coefficients, come under steady loads because of their values being time-invariable for a given wind speed. However, they can be affected by the deck’s rotation, which adds to the effective wind angle of attack. Therefore, an iterative method is generally required to determine the exact nonlinear mean wind effects [¹–³].

Motion-induced loads involve an issue of aeroelasticity. The aeroelasticity of long-span bridge decks has been addressed extensively. Scanlan et al. [⁴–⁶] established the basic methodology of describing aeroelasticity via flutter derivatives, and it has been adopted broadly in bridge aerodynamics worldwide. The flutter-derivative-expressed aeroelastic effects can be realized in both frequency- and time-domain procedures. The frequency-domain method, with the advantage of time-efficiency, has been adopted extensively in aerodynamic stability and buffeting analyses [⁷–¹³]. In comparison, time-domain simulations have an advantage over the frequency-domain simulations in terms of taking into account structural, geometric, as well as aerodynamic nonlinearities, and hence have acquired extensive applications in recent years. Two types of time-domain function, heretofore, have been used to simulate the motion-induced aeroelastic effects for bridge decks: (1) the impulse functions, in together with the Roger rational functions [¹⁴]; and (2) the indicial response functions, or the so-called Wagner-type functions [¹⁵].

Conventionally, mean wind loads have been dealt with in separation with the aeroelastic and the buffeting effects [¹⁶–¹⁸]. This is a rational treatment when the principle of linear superposition is applicable and when the mean wind effects induced nonlinearities in aeroelastic response are negligible. However, when sorts of nonlinearities become notable, then it is worthwhile to employ an integrated simulation where the mean wind loads, buffeting loads, and the motion-related loads are included simultaneously. Superficially, it seems the combination of mean wind loads with the aeroelastic effects can be performed in a straightforward way. In a time-domain procedure, however, an issue arises as regards the incompatibility between the aeroelastic and the mean wind models. This issue, in together with the mean wind effects induced nonlinear aero-elasticity at very high wind speeds, are what going to be addressed in this paper.

Description of aerodynamic loads

Flutter derivatives

Motion-related wind loads developed on bluff bridge decks involve flow separation, and therefore have been denoted conventionally in terms of the so-called flutter derivatives, obtained either from wind tunnel tests or from CFD simulations. Given sinusoidal motions, the self-excited aerodynamic lift and torque per unit length are given as [⁹,¹⁹]:

(1)L_{se}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B\left[K{H}_{\mathrm{1}}^{*}{\displaystyle \frac{\dot{h}}{U}}+K{H}_{\mathrm{2}}^{*}{\displaystyle \frac{B\dot{\alpha }}{U}}+K^{\mathrm{2}}{H}_{\mathrm{3}}^{*}\alpha +K^{\mathrm{2}}{H}_{\mathrm{4}}^{*}{\displaystyle \frac{h}{B}}\right],

(2)M_{se}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}{B}^{\mathrm{2}}\left[K{A}_{\mathrm{1}}^{*}{\displaystyle \frac{\dot{h}}{U}}+K{A}_{\mathrm{2}}^{*}{\displaystyle \frac{B\dot{\alpha }}{U}}+K^{\mathrm{2}}{A}_{\mathrm{3}}^{*}\alpha +K^{\mathrm{2}}{A}_{\mathrm{4}}^{*}{\displaystyle \frac{h}{B}}\right],

where r = air density; U = wind speed; B = reference width; K = Bw/U is the reduced frequency; w = circular natural frequency; H_i^{*}(i = 1~4), A_i^{*}(i = 1~4) are flutter derivatives, h and a are the vertical and torsional displacements; \dot{h}, \dot{\alpha } are the derivatives of h and a with respect to time t, respectively.

18 flutter derivatives in total have been adopted in literatures for full description of the bridge deck aeroelasticity when the drag and lateral motions p and \dot{p} are involved [⁹]. For sake of simplicity and without loss of generality, we leave aside in this paper the aeroelastic drag effects.

Indicial function

Eqs. (1) to (2) are self-excited loads expressed in a mixed time-frequency domain. They are not able to be applied directly in a time-domain analysis. However, the flutter derivatives can be used to obtain a group of indicial functions (IFs), which are applicable to time-domain procedures. The concept of IFs derives from the classic airfoil theory in the description of the transient evolution of the aerodynamic lift due to an abrupt change of the status (velocity or wind angle of attack), as

(3)L(s)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B{C^{\prime }}_L{\alpha }_{\mathrm{0}}\varphi (s)

where

(4)\varphi (s)=\mathrm{1}-a{\mathrm{e}}^{-bs}-c{\mathrm{e}}^{-ds}

is the indicial lift-growth function; {C^{\prime }}_L^{}=d{C}_L/d\alpha; s = Ut/B is a dimensionless time; a, b, c and d are constants. In bridge aerodynamics developed recently, the indicial growth function can be, if desired, altered to a more flexible evolutionary form as

(5)\varphi (s)=\mathrm{1}-{\displaystyle \sum _{\mathrm{1}}^i{a}_i{\mathrm{e}}^{-d_{i}s}},

where a_i, d_i are constants to be identified and d_i>0. A number of authors applied the indicial functions in time-domain flutter analysis or just for the description of aeroelastic effects [¹²,²⁰–²⁴]. Some among them further developed the indicial growth function to a form as

(6)\varphi (s)=a_0-{\displaystyle \sum _{\mathrm{1}}^i{a}_i{\mathrm{e}}^{-d_{i}s}},

where a₀ is also a parameter to be identified.

In virtue of the concept of the indicial functions and the principle of linear superposition, the aerodynamic loads per unit length due to arbitrary motions can be expressed as

(7)\begin{array}{l}{L}_{se}(x,s)=L_{se\alpha }+L_{seh}\\ ={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B{C^{\prime }}_L\left[{\displaystyle {\int }_{-\infty }^s{\varphi }_{L\alpha }(s-\sigma ){\alpha }^{\prime }(x,\sigma )\mathrm{d}\sigma }+{\displaystyle {\int }_{-\infty }^s{\varphi }_{Lh}(s-\sigma ){\displaystyle \frac{h^{″}(x,\sigma )}{B}}\mathrm{d}\sigma }\right]\end{array}

(8)\begin{array}{l}{M}_{se}(x,s)=M_{se\alpha }+M_{seh}\\ ={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}{B}^{\mathrm{2}}{C^{\prime }}_M\left[{\displaystyle {\int }_{-\infty }^s{\varphi }_{M\alpha }(s-\sigma ){\alpha }^{\prime }(x,\sigma )\mathrm{d}\sigma }+{\displaystyle {\int }_{-\infty }^s{\varphi }_{Mh}(s-\sigma ){\displaystyle \frac{h^{″}(x,\sigma )}{B}}\mathrm{d}\sigma }\right]\end{array}

where {\alpha }^{\prime }is the derivative of \alpha with respect to dimensionless time s, h^{″}the second derivative of h with respect to s, and x is the longitudinal coordinate of the bridge deck.

Equating Fourier transforms of (1), (2) and (7), (8) results in the connections between the flutter derivatives and the indicial functions [¹⁵], for example

(9)K(H_{\mathrm{1}}^{*}-i{H}_{\mathrm{4}}^{*})={C^{\prime }}_L[{\varphi }_{Lh}(\mathrm{0})+{{\bar{\varphi }}^{\prime }}_{Lh}],

where the bar on j denotes Fourier transform and i is the imaginary unit. Connections between individual flutter derivatives and the indicial functions can be obtained, for example:

(10)H_{\mathrm{1}}^{*}(K)={C^{\prime }}_L[{\displaystyle \frac{\mathrm{1}}{K}}-{\displaystyle \sum _{\mathrm{1}}^i\frac{a_{Lhi}K}{d_{Lhi}^{\mathrm{2}}+K^{\mathrm{2}}}}],H_{\mathrm{4}}^{*}(K)={C^{\prime }}_L{\displaystyle \sum _{\mathrm{1}}^i\frac{a_{Lhi}{d}_{Lhi}}{d_{Lhi}^{\mathrm{2}}+K^{\mathrm{2}}}},

where a_{fxi}, d_{fxi} are, respectively, the linear multipliers and exponentials of the i^th term of {\varphi }_{fx} (f = L, M; x = h, a).

Rational function

Bucher and Lin et al [¹⁴,²⁵] extended the concept of indicial functions, and introduced the impulse response functions in simulations of the self-excited aerodynamic loads developed on bridge decks, given as

(11)M_{se}(t)=M_{se\alpha }(t)+M_{seh}(t)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}[{\displaystyle {\int }_{-\infty }^t{f}_{M\alpha }(t-\tau )\alpha (\tau )}\mathrm{d}\tau +{\displaystyle {\int }_{-\infty }^t{f}_{Mh}(t-\tau )h(\tau )\mathrm{d}\tau }],

(12)L_{se}(t)=L_{se\alpha }(t)+L_{seh}(t)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}[{\displaystyle {\int }_{-\infty }^t{f}_{L\alpha }(t-\tau )\alpha (\tau )}\mathrm{d}\tau +{\displaystyle {\int }_{-\infty }^t{f}_{Lh}(t-\tau )h(\tau )\mathrm{d}\tau }],

where f_Ma, f_Mh, f_La, f_Lh are impulse response functions. The Roger rational function has been employed to denote the Fourier transforms of the impulse response functions. For example,

(13){\bar{f}}_{Lh}(i\omega )=a_{Lh\mathrm{1}}+a_{Lh\mathrm{2}}\left({\displaystyle \frac{i\omega B}{U}}\right)+a_{Lh\mathrm{3}}{\left({\displaystyle \frac{i\omega B}{U}}\right)}^{\mathrm{2}}+{\displaystyle \sum _{i=\mathrm{4}}^m\frac{a_{Lhi}i\omega }{i\omega +{\displaystyle \frac{d_{Lhi}U}{B}}}},

(13)where a_{Lh\mathrm{1}}, a_{Lh\mathrm{2}}, a_{Lh\mathrm{3}}, a_{Lhi}(i = 4,...m), d_Lhi (d_Lhi≥0; i = 4,...m) are, again, coefficients to be identified. The connection between these coefficients and the flutter derivatives can be established in a similar fashion as those regarding the indicial functions. For example

(14)H_{\mathrm{4}}^{*}(K)={\displaystyle \frac{a_{Lh\mathrm{1}}}{\mathrm{2}{K}^{\mathrm{2}}}}+{\displaystyle \sum _{i=\mathrm{3}}\frac{a_{Lhi}}{\mathrm{2}{d}_{Lhi}^{\mathrm{2}}+\mathrm{2}{K}^{\mathrm{2}}}},H_{\mathrm{1}}^{*}(K)={\displaystyle \frac{a_{Lh\mathrm{2}}}{\mathrm{2}K}}+{\displaystyle \sum _{i=\mathrm{3}}\frac{a_{Lhi}{d}_{Lhi}}{\mathrm{2}{d}_{Lhi}^{\mathrm{2}}K+\mathrm{2}{K}^{\mathrm{3}}}}.

It is worthy of mention that the indicial or rational functions are also able to be experimentally identified directly from in wind tunnel. For techniques in this regard, authors may turn to works of Cao and Sarkar [²⁶], Chowdhury and Sarkar [²⁷], Caracoglia and Jones [²⁸], etc.

The rational functions (RFs) have also been widely employed in the description of aeroelastic effects [¹⁰,²⁹–³¹]. Neither of the two types of function has an obvious advantage over the other as far as the aeroelasticity is concerned. When mean wind responses are involved, however, a significant discrepancy exhibits in their mean properties (or limiting characteristics), which could result in different mean load values, as will be discussed later.

Mean wind loads

Mean wind loads are interwoven with the motion-induced loads when they are integrated in a time-domain procedure. The steady aerodynamic drag, lift, and pitching moment per unit length are expressed generally via steady aerodynamic coefficients:

(15)\bar{D}(x)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B{C}_D[\alpha (x)],

(16)\bar{L}(x)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B{C}_L[\alpha (x)],

(17)\bar{M}(x)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}{B}^{\mathrm{2}}{C}_M[\alpha (x)],

where C_D, C_L, C_M are steady aerodynamic drag, lift, and pitching moment coefficients. They are functions of the effective wind angle of attack \alpha (x), which is the sum of the initial wind angle of attack and the deck’s mean rotation \bar{\alpha }(x). Hence

(18)\alpha (x)={\alpha }_{\mathrm{0}}+\bar{\alpha }(x)

At low wind speeds, the wind-induced deck rotation \bar{\alpha }(x) is small, and the steady wind loads can be approximated with the following expansions:

(19)\bar{D}(x)\approx {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B\left\{C_D({\alpha }_{\mathrm{0}})+{{\displaystyle \frac{\partial C_D}{\partial \alpha }}|}_{{\alpha }_{\mathrm{0}}}\cdot \bar{\alpha }(x)\right\}

(20)\bar{L}(x)\approx {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B\left\{C_L({\alpha }_{\mathrm{0}})+{{\displaystyle \frac{\partial C_L}{\partial \alpha }}|}_{{\alpha }_{\mathrm{0}}}\cdot \bar{\alpha }(x)\right\}

(21)\bar{M}(x)\approx {\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}{B}^{\mathrm{2}}\left\{C_M({\alpha }_{\mathrm{0}})+{{\displaystyle \frac{\partial C_M}{\partial \alpha }}|}_{{\alpha }_{\mathrm{0}}}\cdot \bar{\alpha }(x)\right\}

It is worthy of noting that, as will be seen later in this paper, these one-order approximations could give rise to significant deviations when large torsional deformation involves at high wind speeds.

Incompatibility between aeroelastic and mean wind loads

Mean properties of aeroelastic models

Considering a situation where the structure, after a short period of transient responses, reaches finally a state of wind-induced steady stochastic oscillation, the stochastic responses of the bridge deck can be expressed as

(22)h(x,t)=\bar{h}(x)+\tilde{h}(x,t),

(23)p(x,t)=\bar{p}(x)+\tilde{p}(x,t),

(24)\alpha (x,t)=\bar{\alpha }(x)+\tilde{\alpha }(x,t),

where \bar{h}(x), \bar{p}(x), \bar{\alpha }(x)are the steady components of vertical, lateral, and torsional responses, respectively, and \tilde{h}(x,t), \tilde{p}(x,t), \tilde{\alpha }(x,t) are the corresponding fluctuating components.

The mean values of the motion-induced aerodynamic loads can be obtained by taking mathematical expectations of the load components shown in expressions (7) to (8), as follow (see appendix for detailed process of deduction):

(25)\begin{array}{l}{\overline{L}}_{se\alpha }(x)=E\left[\underset{s\to \infty }{\lim }{L}_{se\alpha }(x,s)\right]\\ ={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B{C^{\prime }}_L\left[{\varphi }_{L\alpha }(\mathrm{0})\overline{\alpha }(x)-\alpha (x,\mathrm{0})-\underset{s\to \infty }{\lim }{\displaystyle {\int }_{\mathrm{0}}^s{{\varphi }^{\prime }}_{L\alpha }(s-\sigma )E[\alpha (x,\sigma )]\mathrm{d}\sigma }\right]\\ ={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B{{\displaystyle \frac{\partial C_L}{\partial \alpha }}|}_{{\alpha }_{\mathrm{0}}}\cdot \overline{\alpha }(x)\end{array}

(26){\bar{L}}_{seh}(x)=E\left[\underset{s\to \infty }{\lim }{L}_{seh}(x,s)\right]=\mathrm{0}

(27){\bar{M}}_{se\alpha }(x)=E\left[\underset{s\to \infty }{\lim }{M}_{se\alpha }(x,s)\right]={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}{B}^{\mathrm{2}}{{\displaystyle \frac{\partial C_M}{\partial \alpha }}|}_{{\alpha }_{\mathrm{0}}}\cdot \bar{\alpha }(x)

(28){\bar{M}}_{seh}(x)=E\left[\underset{s\to \infty }{\lim }{M}_{seh}(x,s)\right]=\mathrm{0}

where E is the operator for taking mathematical expectation. It is noticed that physical meanings of these four mathematic expectations are explicit; that is, they represent the part of aerodynamic loads induced by the mean wind structural deformations. Eqs. (25) to (28) indicate that, when the IF-denoted model is concerned, the mean wind loads developed on the structure are exactly in accordance with the part of steady aerodynamic wind loads resulted from the deck rotation, as the second terms in Eqs. (20) and (21).

In comparison to the IF-denoted model, the RF-denoted model leads to the following mathematic expectations:

(29)\begin{array}{l}{\overline{L}}_{seh}(x)=\underset{t\to \infty }{\lim }{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}\cdot E\left[a_{lh\mathrm{1}}h(x,t)+a_{lh\mathrm{2}}{\displaystyle \frac{B}{U}}\dot{h}(x,t)+{\displaystyle \sum _{i=\mathrm{3}}^m{a}_{lhi}{\displaystyle \underset{-\infty }{\overset{t}{\int }}{e}^{-\frac{d_{lhi}U}{B}(t-\tau )}\dot{h}(x,\tau )\mathrm{d}\tau }}\right]\\ ={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}\left[a_{lh\mathrm{1}}\overline{h}(x,t)+\underset{t\to \infty }{\lim }{\displaystyle \sum _{i=\mathrm{3}}^m{a}_{lhi}{\displaystyle \underset{-\infty }{\overset{t}{\int }}{e}^{-\frac{d_{lhi}U}{B}(t-\tau )}E\left[\dot{h}(x,\tau )\right]d\tau }}\right]={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}{a}_{Lh\mathrm{1}}\overline{h}(x)\end{array}

(30){\bar{L}}_{se\alpha }(x)=E\left[\underset{t\to \infty }{\lim }{L}_{se\alpha }(x,t)\right]={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}{a}_{L\alpha \mathrm{1}}\bar{\alpha }(x)

(31){\bar{M}}_{seh}(x)=E\left[\underset{t\to \infty }{\lim }{M}_{seh}(x,t)\right]={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}{a}_{Mh\mathrm{1}}\bar{h}(x)

(32){\bar{M}}_{se\alpha }(x)=E\left[\underset{t\to \infty }{\lim }{M}_{se\alpha }(x,t)\right]={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}{a}_{M\alpha \mathrm{1}}\bar{\alpha }(x)

It is noted that in this case the mathematic expectations are of no exact physical meanings, and they are mathematically determined in the process of nonlinear fitting for identification of the group of functions.

The mean wind loads expressed in Eqs. (25) and (27) are in essence the same part of aerodynamic loads denoted by Eqs. (20) and (21). Therefore, one has two options to avoid double counting: remove this part of wind loads from the mean wind models and retain it in the aeroelastic models, or vice versa. However, when the RF-denoted models are employed, neither method is able to deal with this part of wind loads correctly due to the dependence of them to the identified rational functions, as shown in Eqs. (29) to (32).

Load combination

In cases of low wind speeds and small deck rotations, the simplest way to avoid double-counting of the deck rotation induced steady aerodynamic loads is to remove this part of loads from the Eqs. (20) and (21); that is, let

(33){\bar{L}}^{*}(x)={\bar{L}}^{*}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B{C}_L({\alpha }_{\mathrm{0}}),

(34){\bar{M}}^{*}(x)={\bar{M}}^{*}={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}{B}^{\mathrm{2}}{C}_M({\alpha }_{\mathrm{0}}),

and the total aerodynamic lift and torque per unit length are then denoted by the sum of three independent parts, including steady, motion-induced, and buffeting loads:

(35)L(x,s)={\bar{L}}^{*}+L_{se}(x,s)+L_b(x,s)

(36)M(x,s)={\bar{M}}^{*}+M_{se}(x,s)+M_b(x,s)

where L_se(x,s) and M_se(x,s) are motion-induced unsteady aerodynamic lift and torque denoted by Eqs. (7) and (8); L_b(x,s) and M_b(x,s) are time-varying buffeting lift and torque, respectively.

(37)L_b(x,t)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B\left(\mathrm{2}{C}_L{\displaystyle \frac{u(x,t)}{U}}+({C^{\prime }}_L+C_D){\displaystyle \frac{w(x,t)}{U}}\right),

(38)M_b(x,t)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}{B}^{\mathrm{2}}\left(\mathrm{2}{C}_M{\displaystyle \frac{u(x,t)}{U}}+{C^{\prime }}_M{\displaystyle \frac{w(x,t)}{U}}\right),

where u(x, t), w(x, t) are the longitudinal and vertical wind fluctuations, respectively.

This approach of load recombination is inapplicable to the RF-denoted model, since the identification-dependent mean values given by Eqs. (29) to (32) have no equality to the additional mean wind load terms induced by deck rotation. This incompatibility determines the RF-denoted model applicable only when mean structural responses \bar{h}(x), \overline{\alpha }(x) are excluded.

Taken the aerodynamic lift for example, conventional mean wind load, buffeting, and aeroelastic models are listed in table 1 in terms of description, mean properties, and nonlinearity. It is noted fundamental contradictions arise from column (II), (III), and (IV), namely between mean wind and aeroelastic loads. The first contradiction is the mean properties of the aeroelastic model are basically included in the mean wind load model 1 (column II), which could result in double counting; the second contradiction is, while the mean wind load 1 being nonlinear, the mean properties exhibited in the self-excited models are inherent linear, which limits their applicability in cases of very high wind speeds where mean-wind-load nonlinearity could be significant. According to table 1, two integrated methods are listed in table 2 in terms of small structural deflection, large structural deflection, and compatibility. It is noted that the first load combination, namely (I) + (II) + (IV), is always incorrect for both small and large structural deflections and results in incompatibility between mean wind and aeroelastic models. The only feasible combination within the frame of conventional bridge aerodynamics is (I) + (III) + (IV). However, it is still inapplicable to cases of very high wind speeds where mean wind deflection induced aeroelastic nonlinearities are non-negligible, and this issue is to be solved in the next section with the separation of the pseudo-steady effects from the aeroelastic time-domain model.

Separation of the pseudo-steady effects

Since the nonlinearity of mean wind loads is unable to be described by the aeroelastic models, it is favorable to remove the mean values from the aeroelastic models, and to deal with the nonlinear mean wind loads separately. In so doing, the mean-wind-load nonlinearities are computed step-by-step in time-domain with Eqs. (15) ~ (17), in the form of load updating according to deck rotations at every time step.

However, numerical techniques are needed to remove the mean wind loads from the time-domain aeroelastic models. Difficulties arise from two aspects. On the one hand, the mean structural responses are unknown in advance; on the other hand, even the mean structural responses are known for a given case, the transient effect resulted from a sudden taking away of the \bar{\alpha }(x) from the aeroelastic model need to be dealt with in an appropriate way. To this end, the following pseudo-steady structural responses are introduced:

(39)\stackrel{︵}{h}(x,s)={\displaystyle \frac{\mathrm{1}}{t}}{\displaystyle {\int }_{\mathrm{0}}^{t}h(x,\tau )\mathrm{d}\tau }={\displaystyle \frac{\mathrm{1}}{s}}{\displaystyle {\int }_{\mathrm{0}}^{s}h(x,\tau )}\mathrm{d}\tau

(40)\stackrel{︵}{\alpha }(x,s)={\displaystyle \frac{\mathrm{1}}{t}}{\displaystyle {\int }_{\mathrm{0}}^t\alpha (x,\tau )\mathrm{d}\tau }={\displaystyle \frac{\mathrm{1}}{s}}{\displaystyle {\int }_{\mathrm{0}}^s\alpha (x,\tau )}\mathrm{d}\tau

It is noted these pseudo-steady responses converge rapidly as time progresses, and one has the following limiting characteristics for them:

(41)\underset{s\to \infty }{\lim }\stackrel{︵}{h}(x,s)=\overline{h}(x),\underset{s\to \infty }{\lim }\stackrel{︵}{\alpha }(x,s)=\overline{\alpha }(x)

These limiting characteristics result immediately in

(42)\underset{s\to \infty }{\lim }{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B{C^{\prime }}_L{\displaystyle {\int }_{-\infty }^s{\varphi }_{L\alpha }(s-\sigma )\stackrel{︵}{{\alpha }^{\prime }}(x,\sigma )\mathrm{d}\sigma }={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B{{\displaystyle \frac{d{C}_L}{d\alpha }}|}_{\alpha \mathrm{0}}\cdot \overline{\alpha }(x)

(43)\underset{s\to \infty }{\lim }{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B{C^{\prime }}_M{\displaystyle {\int }_{-\infty }^s{\varphi }_{M\alpha }(s-\sigma )\stackrel{︵}{{\alpha }^{\prime }}(x,\sigma )\mathrm{d}\sigma }={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}{B}^{\mathrm{2}}{{\displaystyle \frac{d{C}_M}{d\alpha }}|}_{\alpha \mathrm{0}}\cdot \overline{\alpha }(x)

The right sides of the above two expressions are exactly the linear part of mean wind loads result from the deck rotation; therefore, they are able to be subtracted from the aeroelastic model since they are also expressed in terms of the same indicial functions. Finally, the total aerodynamic lift and torque can be rewritten as follow:

(44)L(x,s)=\bar{L}(x,s)+{\tilde{L}}_{se}(x,s)+L_b(x,s)

(45)M(x,s)=\bar{M}(x,s)+{\tilde{M}}_{se}(x,s)+M_b(x,s)

where

(46)\begin{array}{l}{\tilde{L}}_{se}(x,s)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}B{C^{\prime }}_L\left[{\displaystyle {\int }_{-\infty }^s{\varphi }_{L\alpha }(s-\sigma ){\alpha }^{\prime }(x,\sigma )\mathrm{d}\sigma }+{\displaystyle {\int }_{-\infty }^s{\varphi }_{Lh}(s-\sigma ){\displaystyle \frac{h^{″}(x,\sigma )}{B}}\mathrm{d}\sigma }\right]\\ \begin{array}{lll}& & -{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\end{array}\rho U^{\mathrm{2}}B{C^{\prime }}_l{\displaystyle {\int }_{-\infty }^s{\varphi }_{L\alpha }(s-\sigma )\stackrel{︵}{{\alpha }^{\prime }}(x,\sigma )\mathrm{d}\sigma }\end{array}

(47)\begin{array}{l}{\tilde{M}}_{se}(x,s)={\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}{B}^{\mathrm{2}}{C^{\prime }}_M\left[{\displaystyle {\int }_{-\infty }^s{\varphi }_{M\alpha }(s-\sigma ){\alpha }^{\prime }(x,\sigma )\mathrm{d}\sigma }+{\displaystyle {\int }_{-\infty }^s{\varphi }_{Mh}(s-\sigma ){\displaystyle \frac{h^{″}(x,\sigma )}{B}}\mathrm{d}\sigma }\right]\\ \begin{array}{lll}& & -\end{array}{\displaystyle \frac{\mathrm{1}}{\mathrm{2}}}\rho U^{\mathrm{2}}{B}^{\mathrm{2}}{C^{\prime }}_M{\displaystyle {\int }_{-\infty }^s{\varphi }_{M\alpha }(s-\sigma )\stackrel{︵;}{{\alpha }^{\prime }}(x,\sigma )\mathrm{d}\sigma }\end{array}

and

(48){\stackrel{︵}{\alpha }}^{\prime }(\sigma )={\displaystyle \frac{d\stackrel{︵}{\alpha }}{d\sigma }}={\displaystyle \frac{\mathrm{1}}{\sigma }}\alpha (x,\sigma )-{\displaystyle \frac{\mathrm{1}}{{\sigma }^{\mathrm{2}}}}{\displaystyle {\int }_{\mathrm{0}}^{\sigma }\alpha (x,\tau )}\mathrm{d}\tau

Note that the last terms on the right side of Eqs. (46) and (47) remove the linear wind loads induced by the pseudo-steady structural responses, which converge to the mean deck rotation as time progresses.

Numerical results

Details of the FE model employed in the numerical analysis are neglected for short. To check the compatibility between the aeroelastic models and the mean structural responses, the step response of the deck to mean wind loads and the initial nonequilibrium forces is examined. Four different load combinations are considered regarding the mean wind loads and the aeroelastic effects. They are: (1) without aeroelastic effects; (2) RF-denoted aeroelastic effects; (3) with IF-denoted aeroelastic effects (method A); (4) IF-denoted aeroelastic effects but pseudo-steady effects are separated (method B). Concise descriptions of these four patterns of load application are presented in table 3.

What one need to bear in mind is, regardless of whether or not the aeroelastic effects are included, correct structural step response should be damped asymptotically to the same steady state. In another word, the aeroelastic effects should not have any effects on the final position of static equilibrium, or, on the mean structural responses. Fig. 1 plots the mid-span responses to the step loads (mean wind loads and the nonequilibrium static loads). Three points can be observed: (1) At the current wind speed, obvious positive aerodynamic damping is resulted from the aeroelastic effects (see Fig. 1(b)); (2) the response of no aeroelastic effects and the one of IF-denoted aeroelastic effects converge asymptotically to the same limit (static position of equilibrium); (3) the RF-denoted aeroelastic model converge finally in a wrong position.

It is noted that the difference between RF- and IF-denoted functions should be irrelevant to the numerical identification, since flutter derivatives re-obtained from both kinds of functions fit equally well with the original derivatives (see Fig. 2), showing that both models describe accurately the target aeroelasticity. Therefore, instead of from the fitting error, the deviation of the RF-denoted model revealed in Fig. 1 must originate from its inherent non-physical limiting characteristics, as discussed in section 3.1. Further, it is worthy of mention that the quantitative deviation resulted from the RF-denoted model is of no universal meaning, since the mean properties of RFs, denoted with Eqs. (29) to (32), depend on an exact numerical identification.

As discussed in section 3.2, the application of method A should be limited to cases of low-to-moderate wind speeds where mean-wind-load nonlinearity is negligible. At very high wind speeds, to check the applicability of method A and B, again, we use the positions of static equilibrium to which the step responses converge as a standard to evaluate whether the steady aerodynamic nonlinearity is properly included. A wind speed of 90m/s is employed. Three methods are employed to provide comparison, method A, B, and the one without aeroelastic effects.

Time histories in relation to the three methods are presented in Fig. 3. It is noticed that both the vertical and torsional histories corresponding to method B converge asymptotically to the results of no aeroelastic effects. This accordance in static equilibrium position demonstrates the applicability of method B to cases when dynamic/stochastic wind loads are included and the resulted mean structural responses are of importance. In contrast, method A converges gradually to incorrect positions due to its inherent mean-wind-load linearity.

Finally, with the technique of subtraction of the pseudo-steady effects from the aeroelastic model, integrated simulations can be performed where mean wind loads, buffeting loads, and motion-induced aeroelastic effects are simulated simultaneously, as shown in Fig. 4. For comparison, the step response to mean wind loads and the history of pseudo-steady rotation are also plotted. The step response converges to a benchmark, mean of the stochastic oscillation. It can be seen that the pseudo-steady deck rotation, according to which the linear loads are subtracted from the aeroelastic model, virtually overlaps the progressively-damped step response.

Mean wind response induced nonlinearity

In order to investigate the influence of the mean wind response to aeroelastic response, two wind fields are considered, one with a moderate mean wind speed of 30 m/s, another with a high mean wind speed of 80 m/s. As shown Fig. 5, corresponding respectively to these two wind speeds, (a) and (b) are the mean wind torsional responses; (c) and (d) are buffeting responses (aeroelastic effects have been taken into account); (e) and (f) are the algebraic sum of (a) and (c), (b) and (d), respectively, representing conventional treatment of mean wind and buffeting responses; (g) and (h) are integrated simulations resulted from method B. Relations among figures in Fig. 5 are listed in table 4.

It is noticed that at low wind speed, i.e., 30m/s, no significant distinctions can be found between the integrated simulation and the algebraic sum of the mean and the buffeting responses; that is, Fig. 5(e) almost equals Fig. 5(g). In the case of high wind speed 80 m/s, however, significant discrepancies, including not only the signal intensity but also the frequency components, can be found between the two time histories, (f) and (h). This kind of nonlinearity can be illustrated in a clearer way by taking FFT transforms of the time histories. The results are given in Fig. 6, where it is noted that, at the moderate wind speed 30 m/s, the two spectrums, corresponding to the algebraic sum of the mean response plus buffeting and the integrated simulation from method B, are basically in accord with each other, both the curve shape and the amplitude. At high wind speed 80 m/s, in contrast, substantial distinction can be noticed in the low frequency region. In this case, the influence of the mean wind response manifests not only in shifting the predominant buffeting frequency to a lower one, in this example from 0.2 to a point around 0.16, but also in strengthening the response in the whole low-frequency region. This is most likely the result of a substantial weakening of the effective structure stiffness by the negative steady aerodynamic stiffness, of which the mechanism is worthy of further investigation.

Conclusions

In this paper, the incompatibility between the time-domain mean wind and aeroelastic models, as well as the aeroelastic nonlinearities resulted from the mean wind structural responses, are investigated. The following conclusions can be made based on the presented discussions: (1) Mean properties of IF-denoted aeroelastic models have explicit physical meaning, whereas those of RF-denoted aeroelastic models depend on numerical identification and hence are of no specific physical meaning. This leads to the inapplicability of the RF-denoted models to integrated time-domain procedures. (2) At high wind speeds when the steady aerodynamic nonlinearity due to the deck rotation becomes obvious, the IF-denoted model (method A) could give rise to significant deviations. A new method put forward in this work, namely, subtraction of the pseudo-steady effects from the aeroelastic models (method B), can solve successfully both the issues of incompatibility and mean-wind-load nonlinearity. (3) Nonlinearity in the aerodynamic response, which is caused by the mean wind structural responses, becomes significant at very high wind speeds.

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