Gust Buffeting and Aerodynamic Admittance of Structures with Arbitrary Mode Shapes. I: Enhanced Equivalent Spectrum Technique

Consider a structure whose length $l$ is much greater than the reference size $b$ of its cross section; $x$, $y$, $z$ is a local Cartesian system with origin at $o$ (Fig. 1); $z$ coincides with the structural axis, $x$ is aligned with the mean wind direction, and $o$ lies at the height $H$ above ground. Furthermore, $X$, $Y$, $Z$ is a global Cartesian system with origin at $O$; the $X$, $Y$-plane is coplanar with the ground; the $Y$, $Z$-plane is coplanar with the $y$, $z$-plane; $X$ is parallel to $x$; $Z$ is directed upward and passes through $o$; and $z$ is rotated $\varphi$ with respect to $Z$.

The wind loading is schematized as a three-variate (3-V) two-dimensional (2D) stationary Gaussian process, in which the $\alpha =x$, $y$, $\theta$ component is given bywhere $0\le z\le l$; $t$ = time; $f_x$, $f_y$, and $f_{\theta }$ = alongwind force, crosswind force, and torsional moment per unit length; ${\overline{f}}_{\alpha }$ = mean value of $f_{\alpha }$; and $f_{\alpha }^{\prime }$ = zero mean fluctuation of $f_{\alpha }$.

The structure has linear elastic behavior with viscous damping and three uncoupled components of motion, the alongwind displacement $x$, crosswind displacement $y$, and torsional rotation $\theta$. Each $\alpha =x,y,\theta$ generalized displacement is a 2D stationary Gaussian process given bywhere $0\le r\le l$; $\overline{\alpha }$ = mean value of $\alpha$; and ${\alpha }^{\prime }$ = zero mean fluctuation of $\alpha$. The maximum value of $|\alpha |$ in period $T$ over which the mean wind velocity is evaluated is given bywhere ${\sigma }_{\alpha }$, $g_{\alpha }$, and ${\nu }_{\alpha }$ = standard deviation, peak factor, and expected frequency of $\alpha$, respectively; ${\sigma }_{\dot{\alpha }}$ = standard deviation of $\dot{\alpha }$; and ${\mathrm{\Lambda }}_{\alpha }=1$ for $\overline{\alpha }\ne 0$, and ${\mathrm{\Lambda }}_{\alpha }=2$ for $\overline{\alpha }=0$ (Piccardo and Solari 1998a, 2000).

Eqs. (3)–(5) are solved by modal analysis, assuming that damping is small and natural frequencies in each direction are well-separated. Unlike in Piccardo and Solari (1998a, 2000) and following Solari (2018), the quasi-static part of $\dot{\alpha }$ is disregarded. Thuswhere ${\sigma }_{B\alpha k}$ and ${\sigma }_{R\alpha k}=k$th background (quasi-static) and resonant parts of ${\sigma }_{\alpha }$, respectively; $n$ = frequency; $n_{\alpha k}$, ${\xi }_{\alpha k}$, $m_{\alpha k}$, and ${\psi }_{\alpha k}=k$th natural frequency, damping ratio, modal mass, and vibration mode, respectively, in $\alpha$ direction; and ${\overline{f}}_{\alpha k}$ and $S_{f\alpha k}$ = mean value and power spectral density (PSD), respectively, of $k$th modal loading in $\alpha$ directionwhere $S_{f\alpha }(\zeta ,{\zeta }^{\prime },n)$ = cross-PSD (CPSD) of $f_{\alpha }^{\prime }(\zeta ,t)$, $f_{\alpha }^{\prime }({\zeta }^{\prime },t)$; and $\zeta =z/l$ and ${\zeta }^{\prime }=z^{\prime }/l$ = nondimensional coordinates. Expressing the variance of the background displacement as the sum of the modal variances is acceptable only if the contribution of one mode predominates over the other modes. On the other hand, disregard the modal covariances may cause significant errors. This limitation may be overcome by the influence function technique.

Let $\overline{u}$ be the mean wind velocity aligned with $X$, $x$ (Fig. 1); $B_0^2$ are the longitudinal $(X,x)$, lateral ($Y$), and vertical ($Z$) zero-mean turbulence components, where $u^{\prime }/\overline{u}\ll 1$, $v^{\prime }/\overline{u}\ll 1$, and $w^{\prime }/\overline{u}\ll 1$. They are treated here as uncorrelated (Solari and Tubino 2002). The wind loading model adopted by Piccardo and Solari (1996) is simplified by neglecting vortex shedding (Solari 2018), and identifying the wind loading with the gust buffeting. This approach is appropriate provided that vortex shedding occurs at critical mean wind velocities well below the design wind velocity at which gust buffeting is evaluated. In that case, vortex shedding deserves independent evaluations. Accordingly, ${\overline{f}}_{\alpha }$, and $f_{\alpha }^{\prime }$ are defined aswhere $\rho$ = density of air; ${\lambda }_x={\lambda }_y=1$; ${\lambda }_{\theta }=b$; and $c_{\alpha \epsilon }=\alpha$, $\epsilon$th element of $3\times 3$ aerodynamic matrix the rows of which refer to the loading components $\alpha =x,y,\theta$ and the columns refer to the turbulence components $\epsilon =u,v,w$where $c_d$, $c_{\mathcal{l}}$, and $c_m$ = drag, lift, and torsional moment coefficients; $c_d^{\prime }$, $c_{\mathcal{l}}^{\prime }$, and $c_m^{\prime }$ = angular prime derivatives; $I_u={\sigma }_u/\overline{u}$, $I_v={\sigma }_v/\overline{u}$, and $I_w={\sigma }_w/\overline{u}$ = turbulence intensities, where ${\sigma }_u$, ${\sigma }_v$, and ${\sigma }_w$ = standard deviations of $u^{\prime }$, $v^{\prime }$, and $w^{\prime }$; ${\epsilon }^{*}=u^{*}=u^{\prime }/{\sigma }_u$, $v^{*}=v^{\prime }/{\sigma }_v$, and $w*=w^{\prime }/{\sigma }_w$; ${\mathrm{\Sigma }}_{\epsilon }$ = sum of three terms with indices $\epsilon =u,v,w$; and ${\delta }_{\epsilon u}$ = Kronecker’s delta. Because $f_{\alpha }^{\prime }$ is a linear function of ${\epsilon }^{*}$ [Eq. (14)], like ${\epsilon }^{*}$, $f_{\alpha }^{\prime }$ also is a stationary Gaussian process, and its CPSD iswhere $S_{\epsilon \epsilon }^{*}$, $S_{\epsilon }^{*}$, and ${Coh}_{\epsilon \epsilon }$ = CPSD, PSD, and coherence function of ${\epsilon }^{*}$. Using the turbulence model proposed by Solari and Piccardo (2001)where $d_u=6.868$; $d_v=d_w=9.434$; $L_{\epsilon }$ = integral length scale of $\epsilon$ turbulence component in $x$-direction; and $c_{\epsilon }$ = exponential decay coefficient of $\epsilon$ along $z$. Eq. (19) is used in the applications, but may be replaced by any other PSD. Eq. (20) is instead functional to the following developments, but neglects the imaginary part of the CPSD (ESDU 1991), as is typical of all the literature about gust buffeting.

The preceding formulation involves the indices $\alpha$ and $\epsilon$ with the aim of providing the alongwind, crosswind, and torsional actions due to the three turbulence components. If the analysis includes only the alongwind force due to longitudinal turbulence, the indices $\alpha =y$, $z$, and $\epsilon =v$, $w$ disappear, whereas the indices $\alpha =x$ and $\epsilon =u$ can be omitted, making the treatment easier.

Consider the PSD of the $k$th modal wind loading in the $\alpha$ direction. Substituting Eqs. (17) and (18) into Eq. (12), it follows thatwhere ${\overline{\zeta }}_{\alpha k}$ is a suitable value of $\zeta$ (Solari 2018). In the case of horizontal structures (Fig. 1, $\varphi =\pi /2$), the terms evaluated in ${\overline{\zeta }}_{\alpha k}$ are independent of $\zeta$, and Eq. (21) is rigorous.

The generalized equivalent spectrum technique replaces the actual multivariate turbulent field with an equivalent monovariate process, leading to a PSD of the modal wind loading [the first loading in Piccardo and Solari (1998b)] that best approximates the actual PSD. This consists of assuming ${Coh}_{\epsilon \epsilon }=1$ and replacing $S_{\epsilon }^{*}$ with the PSD of the reduced equivalent turbulence

Accordingly, Eq. (21) becomeswhere $F_{\alpha k}$ = participation coefficient; and ${\chi }_{\alpha k\epsilon }$ is a frequency filter that plays the role of an aerodynamic admittance. By virtue of Eq. (20)where $J$ = joint acceptance function; and ${\kappa }_{\alpha k\epsilon }$ = reduced frequency

The aerodynamic admittance coincides with the joint acceptance unless the constant $B_0^2$ is obtained from the conceptualization of the structure as slender linelike in the framework of quasi-steady theory.

The operator $C_0\{h;\eta \}$ and the aerodynamic admittance ${\chi }_{\alpha k\epsilon }$ have the following properties:

In the class of structures with regular modes that do not change sign, $C_0$ can be approximated with excellent precision as (Piccardo and Solari 1998b)

Piccardo and Solari (1998b) evaluated $k$ by an empirical approach that provided results almost coincident with the values furnished by Eq. (30).

The generalized equivalent spectrum technique provides a robust and simplified framework to determine the wind loading and response of slender structures with modes that have constant sign. To extend this formulation to arbitrary modes, the enhanced equivalent spectrum technique herein is formulated. It retains the definition of the PSD of the modal force given by Eq. (23), but replaces Eqs. (24) and (25) with the following expressions:

Unlike $B_0$ [Eq. (24)], $B_{*}$ [Eq. (32)] involves the presence of $|{\psi }_{\alpha k}|$ instead of ${\psi }_{\alpha k}$ and is called the generalized participation coefficient. The advantage of this change is apparent with regard to structures with skew-symmetric modes with respect to $\zeta =1/2$: the modulus prevents the denominator of Eq. (33) from being null and Eq. (23) from being undetermined. Because the new formulation identifies itself with the formulation related to modes that do not change sign, namely $B_{*}=B_0$ and $C_{*}=C_0$, the generalized equivalent spectrum technique may be regarded as a particular case of the enhanced equivalent wind spectrum technique. The definition of ${\overline{\zeta }}_{\alpha k}$ in inclined or vertical structures with complex modes calls for further study.

The operator $C_{*}\{h;\eta \}$ and the aerodynamic admittance ${\chi }_{\alpha k\epsilon }$ have the following properties:

Fig. 2 shows the CFS of Eqs. (32) and (33) given by MATLAB version 2019a software for 17 mode shapes ${\psi }_{\alpha k}$; it also provides ${\chi }_0={\chi }_{\alpha k\epsilon }(0)$ [Eq. (34)], the ${\kappa }_m={\kappa }_{\alpha k\epsilon }$ value for which ${\chi }_{\alpha k\epsilon }$ has a relative maximum, the relative maximum ${\chi }_{\max }$ of ${\chi }_{\alpha k\epsilon }$, and $k^{*}=k_{\alpha k}^{*}$ [Eq. (35)]. Modes are classified into three families (Davenport 1977), Type A, B, and C modes.

Type A modes do not change sign. For Modes 1–5, ${\chi }_0=1$ and ${\chi }_{\alpha k\epsilon }$ monotonically decreases as $n$, $\kappa$ increases following the trend of Eq. (35) [Fig. 3(a)]. For ${\chi }_{\alpha k\epsilon }=C_{*}=C_0$, Eq. (31) provides a CFS of aerodynamic admittance where $k=k^{*}$.

Type B modes are skew-symmetric with respect to the structure midpoint, and, more generally, all modes for which $B_0=0$ [Eq. (24)]. For Modes 6–12, ${\chi }_0=0$, and ${\chi }_{\alpha k\epsilon }$ initially increases as $n,\kappa$ increases, reaches the maximum ${\chi }_{\max }$ for $\kappa ={\kappa }_m$, then decreases with the trend of Eq. (35) [Fig. 3(b)].

Type C modes have intermediate shapes and change sign one or more times with or without a regular pattern. For Modes 13–17, ${\chi }_{\alpha k\epsilon }$ is intermediate between those associated with Type A and B modes [Fig. 3(c)]. ${\chi }_0$ [Eq. (34)] provides a measure of how much a Type C mode approaches a Type A (${\chi }_0=1$) or B (${\chi }_0=0$) mode.

In other words, as noted by Hansen and Krenk (1999), Type A, B, and C modes give rise to different trends of ${\chi }_{\alpha k\epsilon }$ in the low and medium range of ${\kappa }_{\alpha k\epsilon }$; on the other hand, as shown by Eq. (35) and Fig. 3(d), in the high-frequency range, all the ${\chi }_{\alpha k\epsilon }$ diagrams have the same slope when drawn as a function of the logarithm of ${\kappa }_{\alpha k\epsilon }$.

In terms of the aerodynamic admittance growth when the reduced frequency increases, a preliminary but partial interpretation is provided by energy cascade with regard to two elementary structural schemes: a simply supported monospan beam and a simply supported double-span beam. The former has a Type A sinusoidal Mode 2 and the highly coherent large eddies in the low-frequency range cause the maximum displacement. The latter has a Type B sinusoidal Mode 6 in which large eddies cause a self-balanced condition that nullifies displacement. Therefore, it is reasonable that the maximum displacement in a span is attained when eddies acting on that span have a diameter $d$ (proportional to $\overline{u}/n$) comparable with the span length. This interpretation, however, does not clarify the issue of the wind loading on the contiguous span, in which the eddy phase shift plays a key role: if eddies are in phase, displacement is nullified; if eddies act counterphase, they cause the maximum displacement. Unfortunately, energy cascade depicts turbulence in a single point, without considering the phase shift of eddies in different points. This issue was studied by Solari and Martín (2020).

Furthermore, neglect the quasi-steady theory, even in structures with Type A modes may involve aerodynamic admittances greater than unity (Davenport 1961, 1964) due to eddy separation. Several studies debated this issue with regard to bridge deck sections (Diana et al. 2001; Hatanaka and Tanaka 2002; Han et al. 2010). In these cases, however, analyses call for experimental or CFD simulations based on a wide range of methods that often lead to different results, thereby introducing other sources of uncertainties. Furthermore, making recourse to experimental or CFD simulations calls for fixing the structural shape, limiting the generality of the solutions. From this point of view, studying aerodynamic admittance in the framework of quasi-steady theory aims to provide solutions, which at present are not available in literature, that may represent a general starting point for research which disregard quasi-steady theory.

Consider the modal shapewhere ${\overline{\psi }}_{\alpha k}$ does not change sign for $\zeta \in [0,1/2]$; $s=1$ and $s=-1$ correspond respectively to symmetric modes (Type A Modes 1, 2, and 3 in Fig. 2) and skew-symmetric modes (Type B Modes 6, 7, 8, and 9 in Fig. 2) with regard to $\zeta =1/2$ [Fig. 4(a)].

The stretched mode ${\stackrel{⌢}{\psi }}_{\alpha k}(\zeta )={\overline{\psi }}_{\alpha k}(\zeta /2)$ corresponds to stretching ${\overline{\psi }}_{\alpha k}$ [Fig. 4(b)]. Therefore, ${\stackrel{⌢}{\psi }}_{\alpha k}$ does not change its sign for $\zeta \in [0,1]$ and belongs to Type A modes. For example, Modes 2 and 4 are the stretched counterparts of half the Modes 6 and 7, respectively.

Substituting Eq. (36) into Eqs. (32) and (33), these latter becomewhere $D_0$ is an operator defined as

The first term on the right-hand side of Eq. (38) provides the contribution to ${\chi }_{\alpha k\epsilon }$ of the on-diagonal terms in the shaded areas of the integration domain in Fig. 4(c). The second term gives the contribution to ${\chi }_{\alpha k\epsilon }$ of the off-diagonal terms in the unshaded areas. The presence of stretched modes with constant sign in Eqs. (38) and (39) makes $B_{*}=B_0$ and $C_{*}=C_0$.

The operator $D_0\{\stackrel{⌢}{h};\eta \}$ has the following properties:

Fig. 5 shows the CFS of Eq. (39) for Type A Modes 1–5, where $\stackrel{⌢}{h}(\zeta )={\stackrel{⌢}{\psi }}_{\alpha k}(\zeta )\ge 0$. Fig. 6 shows the diagrams of $D_0$, confirming that, whereas $C_0$ has an upper tail similar for any mode shape [Eq. (35) and Fig. 3(d)], the upper and lower tails of $D_0$ depend strictly on the mode. Then, CFS of $D_0$ and $C_0$ [Eq. (31)] are not easy to obtain.

The joint properties of $D_0$ and $C_0$ clarify two remarkable properties of Eq. (38).

For $n=0$, ${\chi }_{\alpha k\epsilon }(0)=(1+s)/2$. Thus, coherently with Fig. 2, ${\chi }_{\alpha \epsilon k}(0)=1$ for $s=1$ (symmetric modes), whereas ${\chi }_{\alpha \epsilon k}(0)=0$ for $s=-1$ (skew-symmetric modes).

For $n$ tending to infinity, the tail of Eq. (38) is given by Eq. (35), in which the $k_{\alpha k}^{*}$ value associated with the stretched mode ${\stackrel{⌢}{\psi }}_{\alpha k}$ is given by Fig. 2 for the pertinent Type A mode. Because Eq. (35) is independent of $s$, the tail of ${\chi }_{\alpha k\epsilon }$ is the same independently of whether the mode is symmetric or skew-symmetric. Similarly to Hansen and Krenk (1999), this aspect also may be interpreted partially by energy cascade and the randomness of high-frequency eddies. A better interpretation was given by Solari and Martín (2020).

Consider an arbitrary mode ${\psi }_{\alpha k}(\zeta )$, where $\zeta \in [0,1]$. It can be expressed aswhere ${\psi }_{\alpha k}^{(e)}$ and ${\psi }_{\alpha k}^{(o)}$ = symmetric (even) and skew-symmetric (odd) parts of ${\psi }_{\alpha k}$ with regard to $\zeta =1/2$, respectively. This representation exists and is unique for any mode. In this section, however, the use of Eq. (42) is limited to modes ${\psi }_{\alpha k}^{(e)}$ and ${\psi }_{\alpha k}^{(o)}$ that do not change sign for $\zeta \in [0,1/2]$ and $\zeta \in [1/2,1]$. For example, the algebraic sum of the Type A symmetric Mode 2 and the Type B skew-symmetric Mode 7 provides a Type C intermediate mode.

Substituting Eq. (42) into Eq. (33), the latter becomeswhere ${\stackrel{⌢}{\psi }}_{\alpha k}^{(e)}(\zeta )={\overline{\psi }}_{\alpha k}^{(e)}(\zeta /2)$ and ${\stackrel{⌢}{\psi }}_{\alpha k}^{(o)}(\zeta )={\overline{\psi }}_{\alpha k}^{(o)}(\zeta /2)$ = stretched counterparts of half the symmetric and skew-symmetric modes [Figs. 4(a and b)], respectively; $F_{\alpha k}$ is the $B_0$ operator [Eq. (24)] applied to Eq. (42); and $F_{\alpha k}^{(e)}$ and $F_{\alpha k}^{(o)}$ are defined as

The joint properties of $D_0$ and $C_0$ clarify two remarkable properties of Eq. (43).

For $n=0$

For $n$ tending to infinity, the tail of Eq. (43) is provided by the first Eq. (35) in whichwhere $k_{\alpha k}^{(e)}$ and $k_{\alpha k}^{(o)}=k_{\alpha k}^{*}$ values associated with stretched modes ${\stackrel{⌢}{\psi }}_{\alpha k}^{(e)}$ and ${\stackrel{⌢}{\psi }}_{\alpha k}^{(o)}$, respectively, provided by Fig. 2 for the pertinent Type A mode.

For example, considering the sum of the Modes 2 and 7 mentioned above, from Eq. (24), $F_{\alpha k}=0.835$; from Fig. 2, $F_{\alpha k}^{(e)}=2/\pi$, $F_{\alpha k}^{(o)}=1/2$, $k_{\alpha k}^{(e)}=4/{\pi }^2$, and $k_{\alpha k}^{(o)}=3/8$. Therefore, ${\chi }_{\alpha \epsilon k}(0)=0.581$, and $k_{\alpha k}^{*}=0.418$.

Eqs. (45) and (46) give some clarification concerning the shape of the aerodynamic admittance of Type C modes [Fig. 3(c)]. For both $n=0$ [Eq. (45)] and $n$ tending to infinity [Eq. (46)], ${\chi }_{\alpha k\epsilon }$ can be interpreted as a weighted average of the ${\chi }_{\alpha k\epsilon }$ functions associated with symmetric Type A and skew-symmetric Type B modes. However, the weights of this average are not constant, but depend on the reduced frequency (Solari and Martín 2020).

Consider an arbitrary mode ${\psi }_{\alpha k}(\zeta )$. The $\zeta \in [0,1]$ domain is separated into a set of subdomains $\zeta \in [{\zeta }_{i-1},{\zeta }_i]$ [Fig. 7(a)], where $i=1,2,\dots ,N\ge 2$ is the order number, and ${\mathrm{\Delta }}_i={\zeta }_i-{\zeta }_{i-1}$ is the length within which the mode is regular and does not change sign. Fig. 7(a) depicts the main situations. According to Fig. 7(b), ${\stackrel{⌢}{\psi }}_{\alpha k}^{(i)}(u)$ is the piecewise mode, where $u=(\zeta -{\zeta }_{i-1})/{\mathrm{\Delta }}_i$ $\in [0,1]$, corresponding to stretch ${\psi }_{\alpha k}(\zeta )$ for $\zeta \in [{\zeta }_{i-1},{\zeta }_i]$; thus, ${\stackrel{⌢}{\psi }}_{\alpha k}^{(i)}$ does not change sign in $u\in [0,1]$, and belongs to Type A modes.

Consider Eq. (26) and evaluate the double integral over the domain $\zeta ,{\zeta }^{\prime }\in [0,1]$ as the sum of the integrals on the subdomains depicted by Fig. 7(c). Eq. (33) may be rewritten

The single sum in Eq. (47) provides the contributions to ${\chi }_{\alpha k\epsilon }$ of the on-diagonal terms in the shaded areas of the integration domain in Fig. 7(c). The double sum provides the contributions to ${\chi }_{\alpha k\epsilon }$ of the off-diagonal terms in the unshaded areas.

The joint properties of $D_0$ and $C_0$ clarify two remarkable characteristics of Eq. (47).

For $n=0$

Thus, ${\chi }_{\alpha k\epsilon }(0)=1$ for modes with constant sign, and ${\chi }_{\alpha k\epsilon }(0)<1$ for modes that change sign.

For $n$ tending to infinity, the tail of Eq. (47) is provided by the first Eq. (35), in whichwhere $k_{\alpha k}^{(i)}=k_{\alpha k}^{*}$ value associated with stretched mode ${\stackrel{⌢}{\psi }}_{\alpha k}^{(i)}$ provided by Fig. 2 for pertinent Type A mode, where $k_{\alpha k}^{*}$ = weighted average of $k_{\alpha k}^{(i)}$ values of its piecewise component Type A modes.

In this framework, aerodynamic admittance can be reconstructed and interpreted as an ensemble of contributions due to the piecewise subdivision of any arbitrary mode into stretched regular modes with constant sign.

The mathematical and conceptual complexity of this matter makes it difficult to obtain a general CFS of the aerodynamic admittance, which is as simple and precise as that derived by Piccardo and Solari (1998a) for Type A regular modes. Despite this, Eq. (47) can be developed in analytical form.

Consider any Type B or C mode, and divide the structural domain $\zeta \in [0,1]$ into $N\ge 2$ subdomains within which the piecewise modes are regular and do not change sign (Fig. 7). Thus, the piecewise stretched modes relating to each $i=1,\dots ,N$ subdomain can be modeled by Type A regular modes. Therefore, substituting the corresponding $C_0=C_{*}$ (Fig. 2) and $D_0$ (Fig. 5) operators into Eq. (47), a rigorous ${\chi }_{\alpha k\epsilon }$ is obtained if the selected Type A modes rigorously match the actual piecewise stretched modes; an approximate solution is found if they approximate the actual modes or the rigorous expression of $C_0$ are replaced by Eq. (25). In any case, the reduced frequency $\kappa$ should be scaled by the appropriate sign and length $\mathrm{\Delta }$.

For example, dividing Mode 6 into two piecewise stretched Modes 2 provides a rigorous solution when using the rigorous expressions of $C_0$ and $D_0$; the solution is approximate when evaluating $C_0$ with Eq. (25). The application of this method to Mode 14 calls for a preliminary approximation described in the section “Structure 2—Steel Chimney.”

Fig. 8 compares the rigorous expressions of ${\chi }_{\alpha k\epsilon }$ given by Eq. (33) for Modes 6, 10, 14, and 15 (solid lines) and Eq. (47) (dashed lines) applied as follows: (1) the actual mode is separated into piecewise stretched regular modes; (2) each of these is approximated by the closest Type A mode; and (3) $C_0$ is evaluated with Eq. (25), whereas $D_0$ is obtained from Fig. 5.

Fig. 8 and other diagrams not reported here show that this method provides excellent approximations with errors that do not exceed a few percent for Type B modes and 10% for Type C modes. In addition, it is easy to apply for modes with a few changes of sign. On the other hand, it becomes laborious for more complex modes and does not allow a clear conceptual interpretation of the link between aerodynamic admittance and the mode shape. Both of these limitations were overcome by Solari and Martín (2020) through POD.

The crosswind fluctuating load in the vertical direction $y$ is due to the longitudinal and vertical turbulence components, $u^{\prime }$ and $w^{\prime }$. Fig. 10 shows the aerodynamic admittances and the PSD of the modal force. Solid lines correspond to the rigorous solution, and dotted lines correspond to the CFS. As is typical of skew-symmetric modes, aerodynamic admittances were null at the zero frequency and their maximum was well below unity. The modal force was due mainly to longitudinal turbulence.

Table 2 lists the main parameters of the vertical displacement for $z=l/4$ due to the first vertical Mode 6. The CFS was applied through Eq. (47) by separating the actual mode into $N=2$ piecewise half-sine modes that rigorously resembled the original mode, and were ${\mathrm{\Delta }}_1={\mathrm{\Delta }}_2=1/2$, $F_{yk}^{(1)}=F_{yk}^{(2)}=1/\pi$, $F_{yk}=2/\pi$, and $k_{yk}^{(1)}=k_{yk}^{(2)}=4/{\pi }^2$.

The CFS provided almost exact results. As is typical of skew-symmetric modes, the background response was small and the resonant response prevailed regardless of structural flexibility.

The alongwind fluctuating load is due to the longitudinal turbulence. Fig. 11 shows the aerodynamic admittance and the PSD of the modal force. Solid lines correspond to the rigorous solution, dotted lines to the CFS. As is typical of intermediate modes, the aerodynamic admittance was well below unity over the whole frequency range, so it quite uniformly eroded the harmonic content of the modal force.

Table 3 lists the main parameters of the alongwind displacement for $z=l$ due to the first longitudinal Mode 14. The CFS was applied by separating the actual mode in $N=2$ piecewise modes that resembled the original mode in an approximate way: the $i=1$ piecewise mode was intermediate between Modes 2 and 3, and the $i=2$ piecewise mode was intermediate between Modes 4 and 5. The $C_0$ and $D_0$ operators in Eq. (47) were evaluated for each piecewise mode as the mean of the related operators, and were ${\mathrm{\Delta }}_1=0.81$, ${\mathrm{\Delta }}_2=0.19$, $F_{xk}^{(1)}=0.217$, $F_{xk}^{(2)}=0.079$, $F_{xk}=0.296$, and $k_{xk}^{(1)}=0.369$, $k_{xk}^{(2)}=0.326$.

Despite the preceding approximations, the CFS gave rise to almost negligible errors.