PROCEDURE AND DERIVATION

F4.4.1 General. The exciting forces on a structure due to wind action tend to be random in amplitude and spread over a large range of frequencies. The structural response is dominated by the action of resonant response to energy available in the narrow bands about the natural structural frequencies. In most cases the major part of the exciting energy is at frequencies much lower than the fundamental natural frequencies of structures and decreases with frequency. Hence for design purposes, with respect to wind loading, it is usually only necessary to consider response in the fundamental modes, as the contribution from higher modes is rarely significant, particularly for the largest values of response. The total response of structures can be classified as being those associated with the hourly mean wind speed or low frequency components and those associated with the gustiness or turbulence of the wind which are predominantly dynamic in character. Davenport (Ref. 30) has discussed the gap in the spectra of strong winds which separates the climatic from the higher frequency gust fluctuations which bring about these two classifications of responses.

As a consequence it has been found convenient to describe forces, moments, deflections, accelerations, etc, in terms of an hourly mean value plus the average maximum likely to occur in an hour which, when added, can then be used as an average hourly maximum, or peak response as it is sometimes called, to define equivalent static design data.

The peak value can be obtained from a probability distribution of the random variables concerned and can be expressed conveniently in terms of the reduced variate, that is the number of standard deviations by which the peak exceeds the mean value. For design purposes it has become practice to define a specific value of this reduced variate and call it a peak factor (g_f) whereby the peak value of a variable (x) can be calculated from Equation F4.4.1.

. . . (F 4.4.1)

^

x x g_fσx

where (^x), (x) and (σx) are the peak, mean and standard deviation values of x respectively, usually related to a record period of one hour for the reasons introduced above. This is discussed in detail by Melbourne (Ref. 31).

The division of response into along-wind and cross-wind is not just a distinction of convenience. The distinction really relates to the forcing mechanisms rather than the response; in fact, the two motions combine to give overall structure response in approximately elliptical paths. The calculations of response are divided into along-wind and cross-wind to accommodate the totally different mechanisms in the calculation models. A proof of the independence of the along-wind and cross-wind mechanisms is given by Melbourne (Ref. 32) who showed, in model and full scale, that the joint probability distributions of the along-wind and cross-wind motions were symmetrical and similar to a bivariate normal distribution.

The wind-induced dynamic response of a slender structure may be substantially increased by the presence of one or more adjacent structures of a similar size (Saunders and Melbourne, Ref. 33; Bailey and Kwok, Ref. 34). The flow around any structure in a group will usually differ from that around a similar isolated structure leading to different forces, both time-averaged and fluctuating.

Interference effects can be divided into these following mechanisms (Ref. 34):

(a) Modification of the incident turbulence mechanism by an upstream structure.

(b) Alteration of the wake excitation mechanism by upstream and downstream structures.

(c) Variations in the quasi-static forces on the structure as it oscillates relative to another (wake flutter and wake galloping).

Interference effects are prevalent in structures located less than 10b apart where b is the dimension of the structure normal to the wind. Structures found in groups, including chimney stacks, multiple pipe runs in chemical plant and tall slender buildings are most affected. Adverse effects may be decreased by geometrical variations in structural shape and relative placement. An example of interference effects on the cross-wind response of a square sectional tower building is given in Figure F4.4.1.

FIGURE F4.4.1 TYPICAL PERCENTAGE CHANGE IN CROSS-WIND RESPONSE OF A SQUARE-SECTION BUILDING B DUE TO A SIMILAR

BUILDING A AT (X, Y)

BUILDING HEIGHT (h) EQUALS 4 BUILDING BREADTHS (b)

F4.4.2 Along-wind response — tall buildings and towers. Since the early 1960s from the work primarily of Davenport (Ref. 35) and Vickery (Ref. 36) it can be concluded that the along-wind response of most structures originates almost entirely from the action of the incident turbulence of the longitudinal component of the wind velocity (superimposed on a mean displacement due to the mean drag).

The gust factor method, as presented in this Standard, is based on a fundamental mode of vibration which has an approximately linear mode shape.

F4.4.3 Cross-wind response — tall buildings and enclosed towers. Cross-wind excitation of modern tall buildings and structures can be divided into three mechanisms (Ref. 32) which are associated with wake; incident turbulence and the cross-wind displacement and its higher time derivatives, which are described as follows:

(a) Wake. For buildings and structures under wind action, the most common source of cross-wind excitation is that associated with ‘vortex shedding’. For a particular structure, the shed vortices have a dominant periodicity which is defined by the Strouhal number. Hence the structure is subjected to a periodic pressure loading which results in an alternating cross-wind force. If the natural frequency of the structure coincides with the shedding frequency of the vortices, large amplitude displacement response may occur and this is often referred to as critical velocity effect. In practice, vertical structures are exposed to a turbulent wind in which both the hourly mean wind speed and the turbulence level vary with height, so that excitation due to vortex shedding is effectively broad-band.

Therefore the term ‘wake excitation’ is used to include all forms of excitation associated with the wake and not just those associated with the critical wind velocity. In this Standard methods for calculating cross-wind response will be restricted to structures under wake excitation.

(b) Incident turbulence. The ‘incident turbulence’ mechanism refers to the situation where the turbulent properties of the natural wind give rise to changing wind speeds and directions which directly induce varying lift and drag forces and pitching moments on the structure over a wide band of frequencies.

The ability of incident turbulence to produce significant contributions to cross-wind response depends very much on the ability to generate a cross-wind (lift) force on the structure as a function of longitudinal wind speed and angle of attack (a). In general, this means a section with a high lift curve slope (dC_l/da) or pitching moment curve slope (dCm/da) such as a streamline bridge deck section or flat deck roof. No methods for calculating cross-wind response for these structures under incident turbulence excitation will be given in this Standard.

(c) Cross-wind displacement. Several cross-wind excitation mechanisms are recognized under the heading of ‘excitation due to displacement’ which more explicitly should read ‘excitation due to cross-wind displacement and higher derivatives of displacement, and rotation’. There are three commonly recognized displacement dependent excitations, ‘galloping’, ‘flutter’ and ‘lock-in’, all of which are also dependent on the effects of turbulence inasmuch as turbulence affects the wake development and hence the aerodynamic derivatives.

(i) ‘Galloping excitation’ results in a single degree of freedom motion which depends on the sectional aerodynamic force characteristics and on the rate of cross-wind displacement to produce a force in phase with the displacement. It is mostly two-dimensional structures such as electrical conductors which are prone to this form of excitation in practice (refer to Parkinson and Brooks, Ref. 37). However, galloping excitation can be significant at very high wind velocities for flexible, lightly-damped and slender tower-like structures (Novak and Davenport, Ref. 38; Kwok and Melbourne, Ref. 39).

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Saunders (Ref.40) discusses the relevance of galloping excitation to typical tall buildings and concludes that galloping and the quasi-steady formulation of negative aerodynamic damping do not have a significant effect on the level of response for V_h/n_cb less than 10. Saunders also concludes that within the range of experiments, which covers many typical rectangular tall buildings, the cross-wind force spectra are insensitive to the level of displacement.

ii) ‘Flutter’ as a name is usually used to cover instabilities and excitations having more than one degree of freedom. In the civil engineering field the bridge deck has suffered and is the one most likely to suffer forms of flutter excitation. A series of studies of flutter in this context by Scanlan and Tomko (Ref. 41) with respect to the effect of various aerodynamic derivatives are of particular interest. The motion of a cantilevered roof in which coupling between rotation and displacement can be significant would also fit partly under a flutter heading.

(iii) ‘Lock-in’ is a term commonly used to describe large amplitude cross-wind oscillations of structure which occur at wind velocities at which the vortex shedding frequencies are close to the natural frequency of the structure. The lock-in excitation mechanism is thought to be due to the cross-wind displacement in which the cross-wind response of a structure causes an increase in the excitation forces, which in turn increases the response of the structure. That is, there is an inter-dependence between the excitation and response processes so that once lock-in becomes established, the vortex shedding frequency tends to couple with the natural frequency of the structure, and the large amplitude response persists. Data applicable to modern tall buildings and structures in natural turbulence boundary layer flow are extremely rare. Wind tunnel model studies by Vickery (Ref. 42), Melbourne (Ref. 43) and Kwok and Melbourne (Ref. 44) are the few reported cases in which lock-in has been found to be significant for tall structures under simulated natural wind conditions. Lock-in is only likely to occur for structures which have relatively low stiffness, are lightly-damped, and are operating near the critical wind velocity (Vcrit) given in Equation F4.4.3.

Vcrit = n_cb . . . (F4.4.3) Sr

where

Vcrit= the critical wind speed, in metres per second

nc = the fundamental natural frequency in the cross-wind direction, in hertz b = the breadth of the structure normal to the wind stream, in metres.

Sr = the Strouhal number

For structures of rectangular cross-section in turbulent flow — Sr = 0.1 for Re > 10⁴

For structures of circular cross-section in turbulent flow — Sr is given in Figure F4.4.3.1, where the Reynolds number Re = 0.67×10⁵V_hb.

In practice the only common structures affected by lock-in are chimney stacks.

The technique employed to calculate the cross-wind response due to wake excitation is to solve the equation of motion for a lightly damped structure in modal form with the forcing function mode generalized in spectral format. The values of the cross-wind force spectrum coefficients given in Figures 4.4.3(A) and (B) are based on a fun damental mode of vibration which has a linear mode shape. Extension of this data to non-linear mode shapes may be obtained by the mode shape correction factor discussed by Holmes (Ref. 45).

The factors (1.06 - 0.06k) and (0.76 + 0.24k) in Equations 4.4.3(1) and 4.4.3(2) are conservative linear correction factors to the base overturning moment and tip acceleration, incorporating corrections to the inertia forces and generalized mass as well as that to the spectrum of generalized cross-wind force.

FIGURE F4.4.3.1 DOMINANT WAKE FREQUENCY ASSOCIATED WITH VORTEX SHEDDING FROM BLUFF BODIES IN TURBULENT FLOW

F4.4.4 Cross-wind response — cantilevered roofs and canopies. The response of a cantilevered roof is dependent on the dynamic response to wind action. This response may be approximately related to the first mode frequency of the cantilevered system as given in Equation 4.4.4. There is obviously dependency on leading edge configurations, and substantial reduction in load can be achieved by using a slotted leading edge. The dependency on mass and damping does not appear to be as much as for many of the other structures in this Section because the response mechanism is not so dependent on the resonance mechanism. (See Melbourne and Cheung (Ref. 46).)

For the design of roof cladding, purlins etc, the static analysis procedures set out in Section 3 should be used.

F4.4.5 Cross-wind response — lattice towers and masts.

F4.4.6 Combination of along-wind and cross-wind responses. As shown by Melbourne (Ref. 68), the dynamic along-wind response and cross-wind response of symmetrical structures each have Gaussian (normal) probability distributions, and are statistically independent of each other. Hence, the joint probability distribution of the along-wind and cross-wind base moment, and their corresponding load effects, is bivariate Gaussian with zero correlation coefficient. It would be conservative therefore, to apply the peak moments ( ^M_a) and ( ^M_c), simultaneously to the structure, as the probability of occurring together, in an hour of wind run at the design hourly mean wind speed under consideration, is much smaller than the probability of occurring separately.

Equation 4.4.6(1) is an excellent approximation to the combined response of scalar structural effects, when the conditions described in the previous paragraph apply, and when the along-wind frequency (n_a) and cross-wind frequency (n_c) are nearly equal to each other (Ref. 69).

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As shown in Figure F4.4.6.1, the combined along-wind and cross-wind dynamic vector response falls within an elliptic envelope. At different times the resultant base moment can take different magnitudes and directions with approximately equal probability. An alternative method to the use of Equation 4.4.6(1) to determine the maximum of any scalar load effect ( ) is to determine it for a range of resultant vector base moments such as those shown in Figure F4.4.6.1 to find the largest ( ).

The peak resultant vector base moment ( ) is equal to the largest distance from the origin to any point on the ellipse. When the mean cross-wind response ( ) is equal to zero, and the dynamic cross-wind response exceeds the along-wind dynamic response (see Figure F4.4.6.2), this distance is given by Equation F4.4.6(1), and the angle (αmax) at which the moment acts is given by Equation F4.4.6(2).

. . . (F 4.4.6(1))

for = 0

> (G− 1)

(F4.4.6(2))

for = 0

> (G− 1) where

= the mean base overturning moment in the along-wind direction G = the gust factor for along-wind response

= the peak base overturning moment in the cross-wind direction.

When the mean cross-wind response ( ) is equal to zero and the cross-wind dynamic response is less than or equal to the along-wind dynamic response, (see Figure F4.4.6.3), the peak resultant base moment is equal to the peak along-wind moment, ( ) equal to ( ).

Similar equations to F4.4.6(1) and F4.4.6(2) can be used for other vector resultants such as acceleration and deflection.

Equation F4.4.6(1) can also be derived by computing the peak base moment ( ) in a plane at an angleα to the mean wind direction, by setting equal to cos α and equal to sin α in Equation 4.4.6(1), and then finding the largest value of for any angleα. This value ( ) is equal to the largest resultant ( ). Note, from Figure F4.4.6.2, that , being a resultant, has no other component in the orthogonal direction, but the largest moment at any other angle ( ) acts together with an orthogonal component in a plane at an angle (90°+ α) to the mean wind direction.

No simple equations like F4.4.6(1) and F4.4.6(2) can be derived for the maximum resultant vector base moment when is not equal to zero, and a graphical or numerical method of solution should be used.

FIGURE F4.4.6.1 RESULTANT VECTOR BASE MOMENTS

FIGURE F4.4.6.2 PEAK RESULTANT VECTOR RESPONSE WHEN M_c= 0, ANDM^

c> (G - 1)M_a

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FIGURE F4.4.6.3 PEAK RESULTANT VECTOR RESPONSE WHEN Mc= 0, ANDM^

c≤(G - 1)Ma

REFERENCES

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