This article is concerned with the prediction of both the along and across-wind response of slender tower-like structures in a turbulent wind. Particular attention is paid to the efficient calculation of the along-wind response and to a model for prediction of the across-wind response to vortex shedding.
Introduction to Response of Tower:
Towers and tower like structures such as chimneys are far more amenable to theoretical analysis than buildings. The prime difference between these two classes of structure is slenderness which enables the tower to be analysed as an essentially one-dimensional structure with aerodynamic properties that are well defined by the local cross-section.
Such a simplification may not be completely adequate in the vicinity of a free end or in the vicinity of significant changes in cross-section but can be accepted in terms of the overall loading. While the one-dimensional rather than three-dimensional aerodynamics is a great simplification the slenderness may introduce more complexity to the dynamic analysis in the sense that many modes of vibration may contribute significantly to the response.
Slenderness is also commonly associated with values of Vh = V̅/foB, [V̅ = a representative mean speed, fo = a representative natural frequency, B = a representative width] which are comparatively high and certainly larger than 1/S [S = Strouhal Number].
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In such cases, the use of quasi-steady aerodynamics is reasonable although in the vicinity of VR = VCR = 1/S there is a need to consider the response to vortex excitation; in addition, at this and higher values of VR, aeroelastic effects (aerodynamic damping etc.) are commonly of importance.
Evaluation of Drag Loads:
The evaluation of the response of multi-degree-of-freedom systems to the random loads induced primarily by atmospheric turbulence is well established. Of particular note is the early work of Davenport [1962] concerning the response of line-like structures and a detailed presentation of the approach through modal analysis due to Harris [1965].
The principal results for a line-like structure are summarized in the following section:
Modal Analysis:
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The equations of motion of the structure in terms of the normal modes are:
MiZi + CiZi + KiZi = Qi(t) …(2.1)
where Zi is the generalized modal co-ordinate and Mi, Ci, Ki and Qi are the modal mass, damping, stiffness and force respectively of the ith mode.
The above formulation implies that the equations of motion can in fact be uncoupled. In reality there will be coupling through the damping of the system. Such coupling is generally weak and commonly ignored but it can be of significance for very closely spaced modes (ωi = ωj) or in systems with unusually strong damping terms.
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The damping coefficient Ci has been expressed in terms of the damping as a fraction of critical. For structures with no specific damping mechanisms the structural damping is normally estimated on the basis of existing data obtained from broadly similar structures but the aerodynamic damping at large values of VR is readily computed.
For motion in the along wind direction the drag force per unit height can be approximated as:
Ignoring second order terms the value of Ci from aerodynamic loading is then:
For a uniform prismatic structure in uniform flow this yields:
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This simple equation (2.3) provides a rapid method of assessing the significance of aerodynamic damping. Using representative values of u̅, fi, D and m the damping can be roughly evaluated and if it appears significant more detailed calculations should be made.
The modal equations may be solved to evaluate the mean values of the modal co-ordinates:
where Ri is the value of R corresponding to unit Zi, i.e.,; the unit mode value of R.
The value of Ri may be evaluated from a knowledge of the influence line and the mode shape from the relationship:
The evaluation of the response statistics is clearly computationally intensive and involves double integration over the structure, integration over frequency and a double summation over the modes. It is for this reason that attention has been paid to simpler approaches which do not significantly compromise accuracy.
The errors introduced by simplification can be quite significant (say 10% or so) without having an appreciable influence on overall reliability since there are physical uncertainties (drag coefficient, structure of atmospheric turbulence, design wind speed, structural properties, damping, etc.) which will normally control the reliability of the overall evaluation. Two such simplified approaches are discussed here, the first is an approach using influence lines to evaluate the low frequency component of the response and the second is the well-known “Gust-factor” approach due to Davenort (1967), Vikery (1971) Simiu (1973) and others.
The Use of Influence Lines:
The influence line approach is based upon the fact that the bulk of the variance in the longitudinal component of turbulence is at frequencies well below the natural (structural) frequencies of vibration and hence the loads induced by those fluctuations can be treated as quasi-static.
If the dynamic force w(z, t) is treated in the quasi-static fashion then the response R(t) can be expressed as:
where IR(z) is the influence function for R i.e., IR(z) is the value of R for a unit load applied at height z.
The mean value of R̅ is then:
And the variance of the low-frequency components of the fluctuating part of the response is:
where Cw(z1, z2) is the covariance of the fluctuating component of the loading at levels z1 and z2.
The resonant response in the vicinity of the natural frequencies can be determined by modal analysis with the added simplification that integration with respect to frequency is not required for lightly damped systems.
The “resonant” contribution to the variance of R is given by:
[SQ,(fi) in the above relationship in the single sided (0 < f < ∞) spectrum of the modal force Qi for the ith mode].
The response, R, is then characterized by the mean value R̅ given by Equation 2.8, and the variance R̅2 given by:
where R2LF and R2RES are given by Equations 2.9 and 2.10 respectively.
Gust Factor Methods:
For many structures the dynamic response is dominated by the lowest mode of vibration and a single-degree-of-freedom approximation is acceptable.
In a true single-degree-of-freedom system the mean response, Z̅1, is:
The expected peak response in some sample of time T, Ẑ1, can be expressed as:
where gT is a peak factor which for a Gaussian narrow-banded response is {Davenport [7]} given by:
where v is the zero-crossing rate and is equal to the square-root of the ratio of the second and zeroth moment of the spectrum of Z.
The gust factor is defined as the ratio of the peak to mean and is given by:
The use of the gust-factor “G” to obtain an estimate of the peak loads is not completely consistent but in most applications it is applied as a multiplier to mean loads or the static response due to these loads. In the case of the American Concrete Institute’s Code for concrete chimney (1988), the gust factor G is applied to the mean base moment but not to the actual mean load distribution.
It is presumed that the loads producing the dynamic base moment, (G – 1) M̅, are distributed in a fashion resembling the distribution of inertia loads. Neither approach is correct and care must be taken when applying the gust-factor approach to certain types of structure. The method can produce very misleading estimates of wind-induced response for some structures e.g., the guyed tower [1985].
Example:
In order to illustrate the three approaches (full modal analysis, influence line approach and the gust-factor method) a simple structure consisting of a vertical, tensioned cable in turbulent uniform flow with characteristics broadly consistent with atmospheric turbulence.
For the purposes of obtaining numerical results the following properties are assumed:
Height of Cable H = 300m
Diameter – D = 0.05m
Mass Ratio (m/ρD2) mR = 4000
Drag coefficient CD = 1.0
Turbulence Spectrum (longitudinal component):
The response studied here is the reaction, R(t), at one end of the cable.
The mean value can be written as:
where F and F2 are dimensionless. Using the influence line approach the value of F is of course equal to the exact value of 0.50.
The modal approach yields:
which, as shown in Fig. 1, leads asymptotically to the value of 0.50. If the first mode only is considered the value is 4/π2 or 0.405; for an accuracy of 2% it is necessary to consider twenty modes of which only the odd modes contribute.
The variance coefficient F2 as computed by the influence line approach is shown in Fig. 2 as a function of the number of modes considered The first term corresponds to the low frequency excitation while the additional increments correspond to the “resonant” peak in each of the modes.
The inclusion of one mode only yields a variance coefficient of 0.197 and the computation converges to a value of 0.202. Similar values are shown in Fig. 3 for the computation by modal analysis.
In this case the coefficient F2 is obtained from the double summation:
After twelve modes the coefficient has attained a value of 0.196 and it is clear that the rate of increase is not negligible. The matrix of the modal contributions F2jk is shown in Table 1.
This table clearly demonstrates that the primary contributions are from the cross-terms F2jk where either j or k is unity. The contribution from these terms is virtually all from the low frequency or quasi-static portion of the spectrum is very inefficiently computed by modal analysis.
If the gust factor approach were applied the factor F would be 0.50 and the value of F2 would be taken as being equal to (F̅2 x F2j/F2j) for j=l.
This yields the result that:
F2 = 0.52 x 0.143/(4/π2)2
= 0.218
Consideration of Non-Linearities:
The analysis in the previous sections is derived from the force/velocity relationship:
In the example the error in the mean is (σu/u̅)2 which is generally small and, for a structure of the dimensions chosen and situated in open country might amount to 1% or 2%.
The inclusion of the non-linear term in the analysis of the fluctuating response will result in errors in both the background response and the “resonant” contributions.
For a Gaussian distribution of u the error in the background contribution to the variance of the local load per unit height is:
The error in standard deviation of the load at a given height is then equal to σ2u/2u̅2 or about 1% in the illustrative example. The distortion of the spectrum due to inclusion of the non-linear terms will have an influence on the resonant response but again this will be small. The errors associated with the linearization of the square law relationship are not large for either the mean or the variance calculations but should not be ignored when considering peak values.
The resonant component of the response will be well represented by a Gaussian distribution but the background or low frequency part is not and the peak factor “g” may be seriously in error if the fluctuating response is due primarily to low frequency loads which are relatively well correlated over the structure.
In the simplest case of a fully correlated load with no resonant effects the relationship between peak load and peak velocity is:
For a typical value of gu, say 3.7, and for the illustrative example with σu/u̅ about 0.12 we have:
gw = 1.2gu
For small structures or rougher terrains in which the typical level of turbulence would be loser to 0.2 the corresponding relationship is:
gw = 1.4gu
The failure to recognize the influence of the square-law relationship on the distribution of the background or low-frequency response can be very significant. The formulation of the Gust factor in the Australian Code [1989] recognizes this potential error and has adopted an empirical adjustment to the peak factor for the background response.
The form adopted can be expressed as:
gR is a peak factor related to the resonant response and R a measure of resonant response.
For a small stiff structure with little resonant contribution B → 1 and R → 0 which yields the result:
Across-Wind Response:
The across-wind (or lift) response of slender tower like structures is induced by the incident turbulence, by the unsteady forces associated with the turbulent wake (i.e., vortex shedding) and by interactions between structural motion and the flow (e.g., aerodynamic damping and stiffness).
Unlike bridge structures, for example, most tower like structures have neither strong lift coefficients nor strong lift coefficient derivatives and hence loads induced directly by either component of the incident turbulence are not likely to play a critical role in design since they will in general be less than the along wind load fluctuations and almost always significantly less than the combined mean and fluctuating components in the along-wind direction.
Even in combination with along-wind loading the across-wind loads due to incident turbulence rarely result in a significant increase in the peak design load for a particular element or cross-section.
The more significant sources of across-wind response are associated with vortex excitation and/or negative aerodynamic damping. Negative damping in a single degree-of-freedom vibration (e.g., galloping) or in a coupled mode (e.g., classical flutter) cannot be ignored but are likely only at speeds well beyond the critical speed and, of course, for cross-sections susceptible to such behaviour. For isolated circular cross-sections only vortex excitation need be considered but other instabilities may be of considerable significance for multiple structures in close proximity.
Vortex Induced Across-Wind Loads:
The forces associated with shedding from a stationary bluff body can be characterized by four aerodynamic parameters, a lift coefficient, a central frequency, a spectral bandwidth and a measure of the span-wise correlation. All four are functions of the cross-sectional shape, the Reynolds Number, the turbulence scale and intensity and, for structures with a height to width ratio less than about twenty, the aspect ratio.
For sharp-edged bodies the parameters can be evaluated by experiment but for the circle and other shapes having curved surfaces the strong dependence on both Reynolds Number and turbulence scale and intensity greatly reduces the value of any model scale measurements and reliance must be made on full scale data which are limited in both quantity and quality.
The following sections are concerned with the definition of the key parameters for circular cross-sections in large scale turbulence and at Reynolds Numbers consistent with full scale structures (generally in excess of 107).
The Spectrum:
The spectrum of the vortex-induced force per unit length is expressed as:
Equation 3.1 defines the spectrum of lift forces induced by shedding and the general form normally provides an adequate fit to measured spectra over the frequency range fs ± Bfs. There is commonly a low frequency component in the spectrum due to forces induced by lateral turbulence but these lateral velocity components have little direct influence on the spectrum near fs or on the maximum response which is induced when fs is near a natural frequency of the structure.
The spectrum is defined by three parameters:
The parameter B depends primarily on the large scale turbulence which produces slow variations in the local velocity. Vickery & Basu [1983] argue that the B should be equal to √B0 + 2i2 where Bo is the bandwidth in smooth flow and i the local intensity of turbulence however limited full-scale data suggest that a simpler relationship.
B = 0.10 + 2i …(3.2)
Is a better predictor for the intensities encountered in atmospheric shear flows. Small scale turbulence, such as produced by a nearby slender structure, will not have a significant effect on B.
The Strouhal Number “S” is dependent upon surface roughness, Reynolds Number, turbulence and the aspect ratio (height/width) of the structure. At Reynolds Numbers beyond about 2×106 there is very little data but S tends to decrease slightly with roughness and increase with turbulence intensity. The dependence on aspect ratio is strong and the suggested relationship is shown in Fig. 4.
This relationship is:
The value of C̅LV has been shown from full-scale measurements, Sanada et al [1992], Weldeck [1992]} to be strongly dependent on the scale and intensity of turbulence. Although the full-scale data indicates this strong dependence on intensity, there is no reason to expect the lift forces coefficient to be changed by large scale turbulence which, in effect, is responsible only for a slowly varying local wind speed. It can be argued that only scales of the same order or less than the diameter could have the observed effects.
This suggests that a modified intensity i* which excludes the very large scales should be employed, such an intensity can be roughly defined as:
i* =i(d/L)1/3 …(3.4)
where L is the integral scale of turbulence.
Using i* as the independent variable the data suggest a relationship;
C̅Lv = [0.15 + 0.55i*] – [0.09 + 0.55i*]e-(20i*)3 …(3.5)
This relationship is shown in Fig. 5 along with data collected from a variety of sources by Vickery & Basu.
In cases where the incident turbulence is not that associated with turbulent atmospheric shear flow (e.g., when the structure is subject to wake turbulence from a nearby structure) the value of i* should be estimated as the intensity of turbulence considering only those scales (wavelengths) less than 10d.
For a chimney which is part of a comparatively closely spaced group all with similar diameters the value of i* will be very similar to the intensity of turbulence within the wake of the upstream structure and may well be beyond the range of i* for which data are available and the fitted relationship [Eq’ n 3.5] may well be in error.
The values of C̅Lv defined by equations 3.4 and 3.5 are for nominally two-dimensional conditions and must be modified to account for aspect ratio (h/d). There is an overall reduction in C̅Lv with decreasing hid and, in addition, the value of C̅Lv reduces rapidly to zero in the immediate vicinity of the tip.
The suggested reduction with aspect ratio is as follows:
Near the tip of the lift coefficient can be further reduced in accordance with the relationship:
Δz = distance from tip; d = tip diameter; β = ½ for circular cross-section.
Cross-Spectra:
Section 3.2.1 defines the spectrum of the vortex-induced loads at a particular cross-section but the relationship between spectra and two cross-sections is also required.
The relationship suggested by Vickery & Basu [1984] is:
In most computations the actual form adopted to define R(z1,z2,f) is not important and it is only the correlation length, l, which is of significance.
Motion-Induced Forces:
The discussion in Section 3.2 concerns the nature of the forces acting on a stationary structure.
If sinusoidal or narrow-band resonant motion is introduced then there are additional forces induced together with changes in the pre-existing forces. The additional forces are motion correlated and comprise a component in phase with the motion (an aerodynamic mass or stiffness force) and an out-of-phase component (the aerodynamic damping, which is commonly negative when the shedding frequency and the frequency of motion are nearly equal).
The forces that existed on the stationary body are also influenced by motion in that span-wise correlation is improved and the spectrum is distorted. While these latter effects are significant they do not play an important role in determining the response of the structure. The changes in spectral distribution and correlation are very noticeable at amplitudes in excess of about five percent of the body width but at these amplitudes the response is determined primarily by the phase related forces, particularly the damping.
The motion correlated forces that develop when the shedding frequency (fs) and the frequency of motion (fv) are roughly equal have been studied by numerous workers [1971, 1969] but the most comprehensive study for tower-like structures was conducted by Stecley [1989, 1993, 1994] who examined both the local and the integrated forces on a square cross-section moving in a base-pivoted mode in turbulent shear flow and the results which follow are derived primarily from this work.
The wind induced base moment may be written as:
where y and y are the tip displacement and velocity respectively and α and β are dimensionless coefficients. In the case of two-dimensional flow the “damping” parameter β is equivalent to mβα/ρB2 where βα is the aerodynamic damping as a fraction of critical and the “stiffness” parameter α is equivalent to -Cm/2 where Cm is the “added mass” coefficient. Although the relationship in Equation 3.9 suggests a linear aerodynamic system both α and β are amplitude dependent. Values of α and β are shown in Fig. 6 and the dependence of these on turbulence level is shown in Fig. 7.
The marked reduction in the magnitude of the negative damping with increasing turbulence is of great practical significance. At turbulence intensities above 20% the negative damping in the vicinity of the critical speed is completely eliminated and it is only at higher reduced velocities that the “negative lift slope” of the square cross-section results in the negative damping associated with galloping.
The aerodynamic damping parameters for a number of simple cross-sections are shown in Fig. 8. All data were obtained from base-pivoted models in shear flow with a “power-law” exponent of 0.11 and a turbulence intensity over the upper part of the model of approximately 8%. The results of Fig. 8 clearly demonstrate the adverse characteristics of the square cross-section.
Conclusion:
Slender structures such as chimneys and towers are amenable to analysis primarily because a comparatively simple aerodynamic approximation (strip theory) yields acceptable results and that quasi-steady assumptions are often but not always acceptable. The present paper attempts to define approaches suited to the prediction of the along-wind or drag response and to provide some data to aid in the prediction of the across-wind response due commonly to vortex excitation.
In the prediction of the along-wind response it is demonstrated that the separation of the computation into a low frequency or background component and a series of modal resonant components is accurate and computationally efficient. The use of conventional modal analysis to evaluate the background response can be extremely inefficient since not only can many modes contribute, the cross-terms between modes are far from insignificant.
In regard to across wind motion the paper outlines a general approach and provides a data base for circular cylinders at high Reynolds Numbers. The question of motion-induced or aeroelastic effects is addressed with examples derived from a study of a prism of square cross-section in turbulent shear flow. The results demonstrate the very important (beneficial) effect of free stream turbulence on the motion induced forces and consequent large amplitude motions.