EACWE 5

Florence, Italy19th – 23rd July 2009

Flying Sphere image © Museo Ideale L. Da Vinci

Keywords: masts, wind profiles, unsteady wind loads, wind-induced vibrations, vortex excitation

ABSTRACT

Frequent wind events with moderate wind velocities are usually crucial for the fatigue of slender structures. In these events, the shape of the wind profiles often differs from common assumptions. In order to examine the effects arising from that, a guyed mast with a circular pole subjected to different realistic wind profiles is analyzed in the time domain. For this, an unsteady transfer model for the buffeting wind loads and, simultaneously, a simulation of the vortex excitation considering lock-in effects are applied. The wind-induced loads are derived from numerically generated, turbulent wind fields, whose properties are verified with full-scale wind data recorded at the 344 m data acquisition tower Gartow. The results of the dynamic analysis show a strong dependency of the stress resultants on the shape of the wind profile. The maximum lateral tip deflection and, hence, the maximum bending moment are obtained for the constant profile, as this profile produces the longest dwell period in the lock-in region.

Contact person: A. Willecke, Technische Universität Carolo-Wilhelmina zu Braunschweig

Institute of Steel Structures, Beethovenstraße 51, 38106 Braunschweig, Germany Phone +49 (0)531 / 391-3375, Fax +49 (0)531 / 391-4592

E-mail a.willecke@is.tu-braunschweig.de

A refined analysis of guyed masts in turbulent wind

M. Clobes, A. Willecke, U. Peil Technische Universität Carolo-Wilhelmina zu Braunschweig

Institute of Steel Structures, Beethovenstraße 51, 38106 Braunschweig, Germany m.clobes@is.tu-braunschweig.de, a.willecke@is.tu-braunschweig.de, u.peil@is.tu-braunschweig.de

1. INTRODUCTION

Wind is the only regularly reoccurring load for slender structures, such as towers or guyed masts. In order to reliably evaluate their fatigue safety, modelling of the frequent wind loads with moderate wind velocities is crucial, because they can produce considerable structural damage.

Particularly in wind events with moderate velocities, the wind properties can differ from commonly used wind profiles. In this paper, the buffeting and the vortex-induced wind loads are therefore derived from numerically generated, turbulent wind fields, whose properties – especially the height-dependent profile of the mean wind velocity – were verified with full-scale wind data.

Common time domain models for buffeting wind loads are based on a quasi-steady transfer of the wind turbulence (Hengst 1999, Sparling & Wegner 2006). However, the aerodynamic admittance is actually frequency-dependent. Especially with respect to low wind velocities this is of particular interest. A refined prediction of the buffeting-induced stress resultants is achieved with an unsteady time domain model, which is also valid for the low wind velocities considered here.

Next to buffeting wind loads, the wind forces acting on structures always contain vortex-induced loads. These are described by a time domain model that combines the description of a mono-component signal in the complex plane (Clobes 2008) and the correlation length model designed by Ruscheweyh (1986). Hence, both the dependency of the amplitude and the vortex shedding frequency on the instantaneous wind velocity and lock-in effects are considered. The intermittent character of vortex excitation is thus taken into account, so that the structure does not necessarily build up its maximum amplitude (Ditlevsen 1985).

2. MODELLING OF NATURAL WIND

2.1 Wind profiles. If the layering of the atmosphere is neutral, the profile of the 10-min mean wind velocity Ū with respect to the height z can be described by a continuous function. In the wind code DIN 1055-4 this is done by the power law according to Hellmann given in Eq. (1). However, the required wind characteristics are generally available in strong winds only.

10ref

zU Uα

⎛ ⎞= ⋅⎜ ⎟⎝ ⎠

(1)

With respect to a fatigue analysis, wind events with moderate wind velocities are usually of interest. In these events, the shape of the wind profile can normally not be described by Eq. (1), as indicated by full-scale wind data, see Fig. 1. In building codes the analysis of vortex excitation is therefore based on the assumption that the critical wind velocity is present at all levels of the structure. This unrealistic assumption has significant influence on structural mode shapes with multiple extrema, e.g. the mode shape of guyed structures or of towers oscillating in higher modes.

The profiles presented in Fig.1 were recorded at the 344m data acquisition tower Gartow. In order to measure the natural wind, cup anemometers and wind vanes are mounted on 17 boom arms that are attached to the West side of the tower from +30 up to +341 m, see Fig. 2(a). The meteorological measuring equipment is shown in Fig. 2(b). Further details on the data acquisition tower are presented in Peil & Nölle (1992), Peil & Behrens (2007) and Clobes (2008).

To evaluate the influence of different profile shapes on the stress resultants due to vortex shedding, multiple time domain analyses with different wind profiles are carried out. For this purpose, the measured profiles are approximated with the simplified profiles (a) to (d) as shown in Fig. 1.

The current wind codes assume a constant variance of the turbulence σu for the height-dependent turbulence intensity Iu (z). Therefore, the profiles Ū (z) and Iu (z) are inversely proportional. The wind data presented in Fig. 1 confirms this visually. Hence, the simplified approach σu = const is used for the numerically generated wind fields, too.

Figure 1: Measured wind profiles at the 344m data acquisition tower Gartow.

Figure 2: 344m data acquisition tower Gartow: (a) boom arms attached to the West side of the tower, (b) meteorological measuring equipment.

2.2 Numerical simulation of turbulent wind fields. The geometric non-linearity is relevant to a structural analysis of guyed structures. In order to take this into account, the structural analysis has to be done in the time domain. For this, a deterministic description of the wind velocity at each structural node is required. The description can be obtained by stochastic simulation methods. With respect to a stationary random process, the simulation can be limited to the turbulent part. The mean values are added afterwards.

The assumption of stationarity is valid for the profiles presented in Fig. 1. If this is generally fulfilled for wind events with moderate wind speeds, will be analyzed in future.

In this paper, the wind fields are generated with the method of wave superposition according to Shinozuka & Jan (1972). The method is based on the complex spectral density matrix Sk ( f ) for all turbulence components k = u, v, w. For u (t) and v (t) the auto power density spectrum of von Kármán and for w (t) the one of Busch & Panofsky is used. The coherence is described by an exponentially decaying approach incorporating height-dependent decay parameters.

The quality of the wind field simulation is shown in Fig. 3. By means of a simple example with three nodes, where the vertical distance of node 1 to node 2 and 3 is 30 m and the horizontal distance between node 2 and 3 is 20 m, this figure indicates a good agreement of the target and the simulated auto power spectral density and coherence.

In order to take the correlation between the nodal forces into account, a joint acceptance function is introduced. This, other parameters as well as further details on the numerical simulation of wind fields are presented in Clobes (2008).

Figure 3: Simulated wind field: (a) auto power spectral density, (b) coherence

3. BUFFETING WIND LOADS IN THE TIME DOMAIN

The buffeting wind loads are described with an unsteady transfer model of the turbulence using aerodynamic impulse response functions IF,k (t) that are convoluted by the instantaneous wind velocity W (t) and the fluctuating wind direction φ' (t). Eq. (2) and (3) show the buffeting drag FW (t) and lift FQ (t), where ρ represents the density of the air, CW the drag coefficient, D the diameter and Li the element length, cp. Fig. 4.

( ) ( )2,

0

( ) ( ) ( ) cos ( )2

t

W W i W uF t C D L I t W dρ τ τ ϕ τ τ′= ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅∫ (2)

( ) ( )2,

0

( ) ( ) ( ) sin ( )2

t

Q W i Q vF t C D L I t W dρ τ τ ϕ τ τ′= ⋅ ⋅ ⋅ ⋅ − ⋅ ⋅∫ (3)

The aerodynamic impulse response functions were derived from full-scale wind data of the turbulent wind and the resulting wind forces, assuming a linear, time-invariant transfer with output

noise only (Peil & Clobes 2008). For the identification of the transfer, the pole of the data acquisition tower Gartow is covered by cladding elements at two the levels +66 and +102 m, where the middle elements rest on load cells, see Fig. 5.

Figure 4: Wind velocities and forces on a circular cross section.

In order to ensure the causality of the transfer system, the complex aerodynamic admittances were approximated in the frequency domain by rational functions. Details on the measurements and the identification of the unsteady transfer model are given in Clobes (2008).

Figure 5: Measurement of the wind transfer at the data acquisition tower Gartow.

4. TIME DOMAIN MODEL OF VORTEX-INDUCED WIND LOADS

4.1 General. In addition to the fluctuating wind velocity and direction, the forces acting on structures are generated by body-induced turbulence that arises from vortex shedding. In this connection, a distinction between fixed bodies and bodies that are free to oscillate has to be made. In case of a fixed body, the mechanism is a purely forced excitation. For a body that is free to oscillate, this changes to an aeroelastic interaction between the structure and the flow, when the velocity of the incident flow is appropriate. The latter is a self-excited oscillation called lock-in.

There are different models to describe vortex-induces wind forces (e.g. Ruscheweyh 1986, Simiu & Scanlan 1996). All models have in common that the influence of forced excitation and self-excitation as well as the transition between both excitation modes is described with empirical parameters derived from wind tunnel experiments. In this paper, the correlation length model applied in DIN 1055-4 is used. Necessary extensions for the time domain analysis in turbulent flow are

described in the following subsections.

4.2 Loads due to forced excitation. Due to the shedding of vortices, an alternating force perpendicular to the flow arises. In general, this force has a narrow-band frequency content concentrated around the Strouhal frequency. In the case of a fixed body in laminar flow, the vortex shedding can therefore be described as a good approximation by a harmonic load with the Strouhal frequency fSt, see Eq. (4), where clat is the aerodynamic force coefficient of vortex excitation. The Strouhal frequency depends on the wind velocity W, the diameter D and the Strouhal number St. In this paper, St = 0.2 is used.

( )2, ( ) sin 2

2Q St lat i St iF t c D L W f t Lρ π= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ (4)

In turbulent flow, the Strouhal frequency is a time-dependent process due to the fluctuating wind velocity, see Eq. (5). In the used model, exactly one frequency fSt (t) is to be related to the amplitude of the vortex-induced force FQ,St (t) at each time step. This kind of signals is called mono-component.

( )( )StW t Stf t

D⋅

= (5)

An extension of Eq. (4) with the time-dependent Strouhal frequency produces a signal whose spectral characteristics are comparable to white noise (Höffer & Niemann 1993). Hence, this is not adequate to generate a time series with only one instantaneous frequency fSt (t) at each time step t.

A process with the instantaneous frequency fshe (t) can be described with a pointer in the complex plane rotating with a fluctuating angular velocity. Vortex-induced wind forces can thus be formulated according to Eq. (6) and (7), where the shedding frequency fshe (t) can be interpreted as fSt (t). In this context the initial phase θ0 has no physical meaning and can be ignored. Similar equations were published by Höffer & Niemann (1993) and D’Asdia & Noè (1998).

( ) ( )2, ( ) ( ) sin ( )

2Q St lat iF t c D L W t tρ θ= ⋅ ⋅ ⋅ ⋅ ⋅ (6)

00

( ) 2 ( ) dt

shet f t tθ π θ= ⋅ +∫ (7)

The spectral characteristics of the presented vortex forces comply with those observed in wind tunnel tests by Vickery (1972), see Fig. 6(a). In addition, the instantaneous frequency is equivalent to the Strouhal frequency fSt (t), Fig. 6(b). This can be proven using Hilbert transforms.

Figure 6: Simulated vortex forces: (a) Comparison of auto power spectral densities, (b) instantaneous frequency.

4.3 Loads due to self-excitation. For a body that is free to oscillate, the vortex excitation is controlled by an interaction between the structure and the flow. As soon as the vortex frequency fSt and the oscillating frequency of the structure fosc are within the frequency range ε0 ⋅ fosc, lock-in occurs. The shedding process is then controlled by the movement of the structure and the Strouhal frequency disappears. As a result, the oscillation amplitude and the correlation of the vortex forces along the axis of the structure increase. There are several lock-in regions with respect to multiple-degree-of-freedom systems.

The maximum oscillation amplitude is limited by the damping and is only reached if the incident wind velocity produces a shedding frequency close to the lock-in range over a long time period, see Fig. 7. However, if the turbulence intensity is high, this range is left quite often (Galemann 1993). As a consequence, the structure does not necessarily build up its maximum amplitude.

Figure 7: Vortex-induced response of a structure in turbulent flow (idealized)

Due to vortex shedding, multiple mode shapes are generally activated. There are several numerical algorithms that allow the instantaneous frequencies to be determined, e.g. Wavelet or Hilbert Huang transforms. However, it is difficult to identify the controlling instantaneous frequency of lock-in, if there are more than one. Therefore, a robust approach is used here, where the actual oscillation frequency fosc (t) is determined by the last two extrema of the oscillation, cp. Fig. 7.

Once the Strouhal frequency fSt (t) is within the lock-in range according to Eq. (8), the excitation frequency fshe (t) is replaced by the current oscillation frequency fosc (t). The width of the lock-in range is modeled by the bandwidth parameter ε0 = 0.3, irrespective of the flow condition. During lock-in, the excitation phase is modified through θ0 in such a way that the vortex force always precedes the structural oscillation y (t) by π/2.

( )( ) ( ) ( ) ( ) ( ) ( )( ) ( ) ( ) ( ) ( ) ( )

0 0

0 0

, 1 2 / 5 , 1 3/ 5

, 1 2 / 5 , 1 3/ 5St St osc osc

sheosc St osc osc

f t f t f t f tf t

f t f t f t f t

ε ε

ε ε

⎧ ∉ − ⋅ ⋅ + ⋅ ⋅⎡ ⎤⎪ ⎣ ⎦= ⎨∈ − ⋅ ⋅ + ⋅ ⋅⎡ ⎤⎪ ⎣ ⎦⎩

(8)

With increasing distance, the correlation of the vortex forces decreases rapidly. Consequently, in each antinode synchronized vortex forces are only applied within an amplitude-dependent correlation length Li = Lk (t) according to DIN 1055-4. In this method, the upper limit Lk (t) / D = 12 takes the decrease of the excitation force at high lateral oscillation amplitudes into account (Ruscheweyh 1986). Outside the correlation length no vortex forces are applied.

The local aerodynamic force coefficient clat is adjusted to the wake condition with respect to the Reynolds number Re. However, the flow condition does not change as fast as the instantaneous wind

velocity. Hence, the Reynolds number is derived with respect to Ū (z). The increase of the rms-value of the force coefficient clat to its maximum value is included in Lk (Ruscheweyh 1986).

4.4 Verification of the time domain model. The analysis of chimneys with a height of 38 to 145 m confirms the presented model for vortex excitation, cp. Table 1. The structural parameters and the measured data were taken from Ruscheweyh & Hirsch (1975) and Ruscheweyh & Verwiebe (1995). Due to the unknown wind characteristics on site, Ū (z) = const = ucrit and Iu (z) = const = 0.1 were assumed for verification.

Table 1: Verification of the time domain model with measured lateral oscillation amplitudes

Lateral oscillation amplitude Height Diameter Eigen- frequency

Critical velocity

Damping Scruton number measured DIN 1055-4 simulated

H D fe ucrit δs Sc ymax ymax ymax [m] [m] [Hz] [m/s] [-] [-] [m] [m] [m] 38 1.016 0.68 3.5 0.030 10.7 0.06 0.07 0.09 99 4.25 0.43 9.0 0.012 3.25 ~ 0.38 0.48 0.52 100 6 0.61 18.3 0.042 3.06 ~ 0.80 0.93 1.2 120 4.9 0.49 12.0 0.018 2.9 0.60 0.65 0.59 145 6 0.5 15 0.031 2.7 1.2 1.0 1.0

5. TIME DOMAIN ANALYSIS OF A GUYED MAST

5.1 General. For the structural analysis, a software is used that allows for the dynamic analysis of geometrically non-linear structures in the time domain under consideration of aeroelastic effects (Clobes 2008). The 205 m guyed mast with a circular pole is analyzed over a period of 600 s. For structural details see Fig. 8. The wind acts parallel to one set of cables. Only buffeting wind loads are applied to the cables.

Figure 8: Structural properties of the mast and chosen mode shapes including the corresponding critical velocity ucrit.

The mast is to be subjected to the four mean wind velocity profiles Ū (z) and their corresponding, inversely proportional turbulence intensity profiles Iu (z) from Fig. 1. Beforehand, the mean wind

velocity and the turbulence intensity of each profile are scaled to the 19th mode shape with ucrit = f19 · D / St = 3.4 m/s. Fig. 9 illustrates the resulting velocity profiles applied to the mast.

Figure 9: (left) Mean wind velocity, (right) U (t) at chosen levels for profile (d).

5.2 Results of the simulation. By means of the tip displacement of the mast subjected to the linear profile (c), Fig. 10 shows the influence of lock-in on the lateral oscillation amplitude. Up to the first half of the 10-min simulation the structure builds up slightly due to resonance in lower antinodes. Not until t = 350 s the shedding frequency fshe (t) equals fosc (t), which results in a steeper increase of the lateral amplitude y. After leaving the lock-in range at the tip at 420 s the oscillation decreases immediately.

Figure 10: Time series of the excitation frequency and tip deflection for profile (c).

With Mx = 290 kNm the constant profile (a) produces the largest bending moment, the maximum lateral tip deflection reaches ymax = 0.28 m, Fig. 11. However, due to the turbulence, the Strouhal frequency momentarily leaves the lock-in region even for the constant profile (a). For the sine (b) and the linear profile (c) the wind velocity in the top part of the structure is mostly above the lock-in range, cp. Fig. 9. Consequently, the maximum bending moment drops to approximately 70% with respect to the constant profile, see Fig. 11. The tip deflection is ymax = 0.19 m and ymax = 0.18 m, respectively. Despite a considerably higher wind velocity at the tip, the stress resultant for the linear profile is slightly smaller than for the sine profile. With respect to the power profile, the bending moment Mx = 230 kNm is 13% larger than the moment of profile (b) and (c). This is the same for the tip deflection.

Figure 11: Lateral bending moments for different wind profiles.

Additional simulations show an increase in the stress resultants by approximately 20% if the buffeting wind loads are omitted (FW (t) = FQ (t) = 0). This is due to the absent lateral buffeting forces FQ (t). Generally, they alternately excite the structure in different directions and hence disturb the harmonic process of building up.

A comparative analysis following the regulations of DIN 1055-4 and using the logarithmic decrement of damping δ = 0.012 shows a maximum tip deflection of ymax = 0.34 m. This amplitude, which can be interpreted as the steady state, is higher than the amplitude resulting for the constant profile (a), because the method of DIN 1055-4 does not explicitly include leaving of the lock-in

region, different periods of lock-in in different antinodes and time that is needed for building up. As expected, the frequency content of the lateral tip deflection y (t) is narrow-banded and

concentrated around the eigenfrequency f19 = 0.57 Hz, see Fig. 12. Other dominant eigenfrequencies, as they are present in the x-direction, are not excited in the lateral direction for the applied wind profiles. As expected, the longitudinal oscillation x (t) is stochastic, cp. Fig. 10.

Figure 12: Auto power spectral density of the tip deflection.

6. CONCLUSIONS AND FUTURE PROSPECTS

A guyed mast subjected to turbulent wind with moderate wind velocities and profile shapes that are derived from full-scale wind data was analyzed. The analysis was conducted in the time domain considering structural non-linearities as well as unsteady buffeting wind loads and vortex-induced lateral wind loads.

The results indicate a strong dependency of the stress resultants on the applied wind profile, with the maximum lateral tip deflection for the constant profile. For the linear profile, the stress resultants are the least. For most of the profiles, lock-in seldom occurs simultaneously in all antinodes. Even for the constant profile, the Strouhal frequency momentarily leaves the lock-in region. This indicates the decisive influence of the bandwidth parameter. In consequence, the steady state is not reached for the applied wind profiles.

So far, the resulting damages could not be calculated, because the probability of occurrence of the different profile shapes within the design period is unknown. In the near future, the probability of occurrence of classified wind profiles and their characteristic wind velocities will be determined on the basis of extensive full-scale wind data of the 344 m data acquisition tower Gartow. Furthermore, characteristics of wind events with moderate wind velocities will be examined.

ACKNOWLEDGEMENTS

We gratefully acknowledge the financial support of the Deutsche Forschungsgemeinschaft (German Research Foundation).

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