Equivalent static wind loads on tall buildings.pdf

BBAA VI International Colloquium on:Bluff Bodies Aerodynamics & Applications

Milano, Italy, July, 20–24 2008

EQUIVALENT STATIC WIND LOADS ON TALL BUILDINGS

Stefano Pasto?†, Luca Facchini†, Lorenzo Procino†, Paolo Spinelli†

†CRIACIV/Dipartimento di Ingegneria Civile e Ambientale Universita di Firenze, via S. Marta 3,50133 Firenze, Italy

e-mails: [email protected], [email protected],[email protected], [email protected]

Keywords: Random Vibrations, Tall Buildings, Wind-Induced Loads, Wind-Exposed Struc-tures; Wind Tunnel

Abstract. The results of experimental campaigns on two tall buildings are presented. Testshave been carried out in the CRIACIV boundary layer wind-tunnel by pressure-tap measure-ments and aerodynamic-balance tests, on both models. Numerical studies have been performed,afterwards, to define the equivalent static background and resonant wind-induced loads consid-ering the structure three-dimensionality. The multivariate extreme theory has been adopted topropose design wind loading combinations.

1 INTRODUCTION

Among the tall buildings already built in Italy, those studied herein are of great concernwithin wind-exposed structural design as, most probably, they are going to be the tallest Ital-ian buildings. So, a careful design against wind-induced actions is a must in these cases. Inparticular, the subjects of the present paper are the towers of UNIPOL (125m tall) located inBologna, and Piazza Garibaldi-Repubblica (140m tall) located in Milano. The towers have beentested in the CRIACIV boundary-layer wind-tunnel by means of pressure-tap measurements andaerodynamic-balance tests.

2 EXPERIMENTAL TESTS

The typical mean wind profile of an urban zone (β = 0.2274 within the exponential pro-file) has been reproduced in wind tunnel, as well as the atmospheric turbulence, within thewell-known approximations in simulating real turbulence length scales. Moreover, the towersmodels and the surrounding urban contexts have been reproduced and placed in the text-sectionof the wind-tunnel, as shown in Fig. 1. In particular, both the models are 1:350 scaled andequipped by 125 and 140 pressure taps in the case of the Unipol and Garibaldi-Repubblicatowers, respectively (see Fig. 2). Pressure fields have been logged within 16 wind directions,equally spaced by 22.5 degrees, by a sampling frequency of 250Hz during 30 seconds.

1

[email protected]�




Stefano Pasto, Luca Facchini, Lorenzo Procino and Paolo Spinelli

UNIPOL tower Garibaldi-Repubblica tower

Figure 1: Showing of the wind-tunnel samples

UNIPOL tower Garibaldi-Repubblica tower

Figure 2: Showing of the pressure-tap connections

2.1 Wind-induced pressure fields

For each pressure tap, the mean and Gumbel extreme values (Ref. [1]) of the pressure coeffi-cients have been computed. The extremes values (maximum and minimum values) belong to areturn period of 50 years. In general, mean and extreme pressure coefficients should be agreedupon as design values.Some pressure maps for both towers are shown in Figs. 3, 4 and 5.

2


Mean Cp Mean Cp

Figure 3: Mean pressure maps at wind angle of attacks α = 0o and α = 11o for the Garibaldi-Repubblica tower (left) and the Unipol tower (right), respectively

Gumbel Maximum Cp Gumbel Maximum Cp

Figure 4: Maximum pressure maps at wind angle of attacks α = 0o and α = 11o for theGaribaldi-Repubblica tower (left) and the Unipol tower (right), respectively

In general, however, neither the mean nor the extreme maps, in Figs. 3, 4 and 5, can beused as design pressure maps. In fact, the mean values alone let just the assessment of the staticstructural response, whereas the extreme maps, although they take into account the fluctuatingcomponent of the wind-induced pressures, might lead to a structural design too much conser-vative as the pressures themselves are not fully correlated among each other in time and space.More reliable maps could be obtained by computing design maps within a multivariate extremetheory, so to have sets of pressure coefficients showing themselves by a given return period.Nevertheless, since, for the sake of good outcomes, pressure fields are supposed to be loggedby several pressure taps, analyses based on the multivariate extreme theory might be incrediblyonerous compared to other fields of application. Moreover each structural response should becomputed for each set of pressure coefficients carried out by this theory, in order to obtain theworst situation. On the other hand, it is possible to use this theory reducing drastically the num-ber of time histories being processed by referring to the base resultant moments and torque, as

3


it is being proposed later on.From the foregoing it follows that those extreme maps might not be reliable for designing wind-exposed structure. So, other procedures might be by suitable. For instance, by computing acertain response first and then defining the design pressure maps for the same response.In the next section the design maps, providing the extreme values of the base resultant forces,are being presented.

Gumbel Minimum Cp Gumbel Minimum Cp

Figure 5: Minimum pressure maps at wind angle of attacks α = 0o and α = 11o for theGaribaldi-Repubblica tower (left) and the Unipol tower (right), respectively

2.2 Design wind-induced pressure maps for base forces

By using the time histories of wind-induced forces it is possible to compute a certain struc-tural response by means of its influence function both by quasi-static and dynamic fields.The pressure fields of both towers have been integrated to obtain the resultant forces acting atthe structure bases. By considering the base resultant forces as the structural responses underexamination, it has been possible to map the pressure fields at the time instant supplying theextreme design values of the respective forces. Each of those extreme values has been foundindependently, that is by neglecting common threshold crossings. The results are reported inFigs. 6. The above design maps should be agreed upon as quasi-static ones, as they derive fromthe integration of the pressure fields logged on rigid wind-tunnel models. In general, the equiv-alent static background wind loads can be obtained by distributing the base moments and torqueto each floor as proposed in Ref. [2], [3] and [4].The equivalent static wind loads for sway motions can be computed following the procedure

proposed in Ref. [5], later on discussed. Briefly, this procedure let the assessment of the equiv-alent static resonant wind-loads starting from the time-histories of the base resultant momentsand torque. Since the structure motion is considered, the base resultants must show frequencycontents in a frequency band involving the structure eigenfrequency at full scale. In order toguarantee this, it is necessary to log the pressure maps by an opportune sampling frequency,according to the similitude criteria necessary for simulating real problems at model scale (seee.g. Ref. [3]).Nevertheless, despite the sampling frequency of the pressure maps was quite high (250Hz), itwas not enough to make the base forces have frequency contents in the desired band. So, thebase resultant forces were also logged by means of aerodynamic-balance tests.

4


Map for maximum Fx at α = 157.50o Map for maximum Fx at α = 11o

Map for maximum My at α = 157.50o Map for maximum My at α = 11o

Map for minimum Fy at α = 247.50o Map for minimum Fy at α = 67.50o

Map for maximum Mx at α = 247.50o Map for minimum Mx at α = 67.50o

Map for maximum Mz at α = 315.00o Map for minimum Mz at α = 67.50o

Figure 6: Design pressure maps providing the design values of the base resultant force. Left)Garibaldi-Repubblica tower. Right) UNIPOL tower

5


Figure 7: Aerodynamic balance

3 AERODYNAMIC BALANCE TESTS

The base resultant forces and moments (along-wind and across-wind forces, moments andtorque) of both towers have been logged by means of an aerodynamic balance. The 5-componentsaerodynamic balance is shown in Fig. 7. Tests have been carried out by means of a samplingfrequency of 1000Hz (4 times the frequency used for logging the pressure maps).Comparisons between the base forces obtained by integrating the pressure fields, and by theaerodynamic balance, are reported in Figs. 8, 9 and 10. In those figures, forces and moments

0 50 100 150 200 250 300 350−1.5

−1

−0.5

0

0.5

1

1.5

2x 10

5

α (°)

Mz (

m3 )

Mz vs. α

0 50 100 150 200 250 300 350−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1x 10

5

α (°)

Mz (

m3 )

Mz vs. α

Figure 8: Base resultant pseudo-torques varying the wind angle of attack (α). Left) Garibaldi-Repubblica tower. Right) Unipol tower. Legend: N: Aerodynamic-Balance tests;E: Pressure-field integrations

should be understood as pseudo-forces and pseudo-moments as their measure units disagreewith the real ones. To switch to the full scale forces and moments, it is just necessary to multi-ply those quantities by the dynamic pressure at the top of the towers, at full scale.It is dutiful to remember that pressure fields provide an approximation of real continuous fieldsas the number of pressure taps is finite and each tap belongs to a certain influence area. Onthe contrary, base forces obtained by aerodynamic-balance tests are much more reliable as theybelong to the continuous wind-induced pressure fields. So, differences may show themselves

6


in comparing the forces obtained within the two procedures. In particular, such differencesare more marked for the Garibaldi-Repubblica tower than for the Unipol tower. This mighthighlight how relevant could be not only the aerodynamic of the structure itself, but also theinfluence of the urban context: in the case of the Garibaldi-Repubblica tower, the urban contextis constituted by taller and closer buildings than the ones surrounding the Unipol tower.As mentioned above, the base forces can be used to obtain the equivalent static wind loadstaking into account structure resonances. The procedure is shown in the next section.

0 50 100 150 200 250 300 3500

1000

2000

3000

4000

5000

6000

7000

8000

9000

10000

α (°)

Fx (

m2 )

Fx vs. α

0 50 100 150 200 250 300 3500

1000

2000

3000

4000

5000

6000

α (°)

Fx (

m2 )

Fx vs. α

0 50 100 150 200 250 300 350−1

0

1

2

3

4

5

6

7

8x 10

5

α (°)

My (

m3 )

My vs. α

0 50 100 150 200 250 300 350−0.5

0

0.5

1

1.5

2

2.5

3

3.5

4x 10

5

α (°)

My (

m3 )

My vs. α

Figure 9: Base resultant pseudo -forces and -moments varying the wind angle of attack (α).Left) Garibaldi-Repubblica tower. Right) Unipol tower. Legend: N: Aerodynamic-Balancetests;E: Pressure-field integrations

7


0 50 100 150 200 250 300 350−8000

−6000

−4000

−2000

0

2000

4000

6000

α (°)

Fy (

m2 )

Fy vs. α

0 50 100 150 200 250 300 350−5000

−4000

−3000

−2000

−1000

0

1000

2000

3000

4000

α (°)

Fy (

m2 )

Fy vs. α

0 50 100 150 200 250 300 350−6

−4

−2

0

2

4

6x 10

5

α (°)

Mx (

m3 )

Mx vs. α

0 50 100 150 200 250 300 350−3

−2

−1

0

1

2

3x 10

5

α (°)

Mx (

m3 )

Mx vs. α

Figure 10: Base resultant pseudo- forces and moments varying the wind angle of attack (α).Left) Garibaldi-Repubblica tower. Right) Unipol tower. Legend: N: Aerodynamic-Balancetests;E: Pressure-field integrations

4 DESIGN WIND-INDUCED QUASI-STATIC AND RESONANT LOADS

The equivalent static resonant wind loads, at height z above ground for sway motions asso-ciated to a certain mode shape, can be computed by (see Ref. [5])

Pr(z) = Mrm(z)φi(z)∫ H

om(z)φi(z)zdz

(1)

for bending moments and

Tr(z) = TrI(z)θj(z)∫ H

oI(z)θj(z)dz

(2)

for torque. In Eqs. (1, 2), m(z) and I(z) are the structure mass and mass moment of inertiafor unit height, respectively; φi(z) and θj(z) are the generic eigenmodes associated to bend-ing and torque deformations, respectively. The resonant moments, MR, and torque, TR, areestimated amplifying the quasi-static moments by the structure transfer function at the bending

8


eigenfrequency, fi, and torque eigenfrequency, fj , respectively:

Mr = gri

√π

4νi

fiSM(fi) Tr = grj

√π

4νj

fjST (fj) (3)

where νk (k = i, j) is the damping ratio associated to the considered mode shape; SM andST are the quasi-static spectra of base moment and torque, respectively; gr =

√2 ln fkt +

0.5772/√

2 ln fkt (k = i, j) is the resonant factor; t is the observation time.The quasi-static spectra mentioned above usually are wide-band spectra. By virtue of this, mo-ments and torque might show frequency contents at the structure eigenfrequencies, so contribut-ing to amplify the structure response. In this case, MR and TR, in Eq.(3), must be evaluated atthe same frequency (e.g. fi for the mode φi(z)) and then projected on the respective componentsof the ith mode shape under consideration. Nevertheless, if the eigenmodes are uncoupled, thatis they just show deformations along a certain direction (x, y or z), the definition of the equiv-alent static wind loads in Eq.(2) only applies to the relevant moment or torque which yieldsdeformation along the eigenmode direction.In the latter case, the equivalent static wind-induced loads, including quasi-static (Pb,x(z),Pb,y(z) and Tb,z(z)) and resonant components (Pr,x(z), Pr,y(z) and Tr,z(z)), may be writtenas following:

Px,tot(z) = Pb,x(z) + Pr,x(z) = Pb,x(z) + gri

√π

4νi

fiSMy(fi)m(z)φi(z)∫ H

om(z)φi(z)zdz

Py,tot(z) = Pb,y(z) + Pr,y(z) = Pb,y(z) + grj

√π

4νj

fjSMx(fj)m(z)φj(z)∫ H

om(z)φj(z)zdz

Ttot(z) = Tb(z) + Tr(z) = Tb(z) + grk

√π

4νk

fkFSMz(fk)m(z)φk(z)∫ H

om(z)φk(z)zdz

(4)

where φi(z), φj(z) and φk(z) are the mode shapes related to the eigenfrequencies fi, fj and fk,mainly developing along x, y and around z, respectively.The quasi-static loads are obtained by integrating the wind-induced pressure fields. In particu-lar, the following procedure have been adopted. For instance, in order to obtain Pb,x(z) we areinterested to find the pressure map corresponding to quasi-static design (Gumbel) value of thebase resultant My at the wind angle of attack αcr = α(My). So, the time instant (te), at whichthe time-history My(t) crosses the Gumbel threshold (by positive or negative slopes accordingto the sign of the design value) is searched. Then, the pressure map belonging to te is integratedalong x at each floor, assuming full scale structure by rigid slabs. Since structures are 3D, wecan also obtain, by the pressure map under examination, Pb,y(z) and Tb,z(z), related to Mx andMz at αcr = α(My), respectively. Clearly, in general, Pb,y(z) 6= Pb,y(z) and Tb(z) 6= Tb(z).The respective resonant components of those loads have been computed by the procedure de-scribed above, so getting the total wind-induced loads according to Eq.(4) in the case of My.The procedure has been repeated for Mx and Mz. In this way, the loads for each αcr (at whichmoments and torque approach their design values; see Fig. 6) have been found. The results arereported in Figs. 11, 12 and 13 for the Garibaldi-Repubblica tower. In particular, the resonantforces and torques have been computed by means of the quasi-static base resultants obtained byboth pressure-map integrations and aerodynamic-balance tests. Those loads are further differ-ent. This is ascribed to different frequency contents of moments and torque spectra, within thetwo procedures.

9


50 100 15020

40

60

80

100

120

140

Px,b

(z) [kN/m]

z [m

]

0 0.5 120

40

60

80

100

120

140

Px,r

(z) [kN/m]50 100 150

20

40

60

80

100

120

140

Px,tot

(z) [kN/m]

(a) Px,b(z), Px,r(z) and Px,tot(z)

0 20 4020

40

60

80

100

120

140

Py,b

(z) [kN/m]z

[m]

0 0.5 120

40

60

80

100

120

140

Py,r

(z) [kN/m] Py,tot

(z) [kN/m]0 20 40

20

40

60

80

100

120

140

(b) Py,b(z), Py,r(z) and Py,tot(z)

−600 −400 −200 020

40

60

80

100

120

140

Tb(z) [kNm/m]

z [m

]

−150 −100 −50 020

40

60

80

100

120

140

Tr(z) [kNm/m]

−600 −400 −200 020

40

60

80

100

120

140

Ttot

(z) [kNm/m]

(c) Tb(z), Tr(z) and Ttot(z)

Figure 11: Garibaldi-Republica tower: equivalent static wind loads at α = 247.5o (Mx = Mx).Legend: · · · resonant loads obtained by aerodynamic-balance time histories

30 40 50 6020

40

60

80

100

120

140

Px,b

(z) [kN/m]

z [m

]

0 0.5 120

40

60

80

100

120

140

Px,r

(z) [kN/m]20 40 60 80

20

40

60

80

100

120

140

Px,tot

(z) [kN/m]


−100 −50 020

40

60

80

100

120

140

Py,b

(z) [kN/m]

z [m

]

−1 −0.5 020

40

60

80

100

120

140

Py,r

(z) [kN/m] Py,tot

(z) [kN/m]−100 −50 020

40

60

80

100

120

140


0 500 1000 150020

40

60

80

100

120

140

Tb(z) [kNm/m]

z [m

]

0 50 10020

40

60

80

100

120

140

Tr(z) [kNm/m]

0 500 1000 150020

40

60

80

100

120

140

Ttot

(z) [kNm/m]


Figure 12: Garibaldi-Republica tower: equivalent static wind loads at α = 157.5o (My = My).Legend: · · · resonant loads obtained by aerodynamic-balance time histories

20 40 60 8020

40

60

80

100

120

140

Px,b

(z) [kN/m]

z [m

]

0 0.5 120

40

60

80

100

120

140

Px,r

(z) [kN/m]20 40 60 80

20

40

60

80

100

120

140

Px,tot

(z) [kN/m]


0 20 40 6020

40

60

80

100

120

140

Py,b

(z) [kN/m]

z [m

]

0 0.5 120

40

60

80

100

120

140

Py,r

(z) [kN/m] Py,tot

(z) [kN/m]0 20 40 60

20

40

60

80

100

120

140


−2000 −1500 −1000 −50020

40

60

80

100

120

140

Tb(z) [kNm/m]

z [m

]

−150 −100 −50 020

40

60

80

100

120

140

Tr(z) [kNm/m]

−2000 −1500 −1000 −50020

40

60

80

100

120

140

Ttot

(z) [kNm/m]


Figure 13: Garibaldi-Republica tower: equivalent static wind loads at α = 315o (Mz = Mz).Legend: · · · resonant loads obtained by aerodynamic-balance time histories

10


5 STATISTICAL COMBINATION OF DESIGN WIND-INDUCED LOADS

For the sake of clarity, let’s keep considering My. The quasi-static 3D-loading, at αcr =

α(My), may be written as follows:

Pb(z)|α(My) = Pb,x(z)i1 + Pb,y(z)i2 + Tb(z)i3 (5)

where i1, i2 and i3 are the versors along the Cartesian coordinates x, y and z. The total load inEq.(5) may be still re-written by introducing the following notation:

Pb(z)|α(My) = γ11Pb,x(z)i1 + γ12P∗b,y(z)i2 + γ13T

∗b (z)i3 (6)

where P ∗b,y and T ∗

b are the loads reproducing to the Gumbel values of Mx(t) and Mz(t) at α(My),that is M∗

x and M∗z So, in order to get the latter loads, it is necessary to integrate the maps at the

instant during which Mx(t) = M∗x and Mz(t) = M∗

z at α(My). The coefficients γ11, γ12 andγ13 have to be understood as coefficients to combine Pb,x(z), P ∗

b,y(z) and T ∗b (z) statistically, as

Mx(t), My(t) and Mz(t) are not fully correlated among each other and hence, generally, they donot approach their Gumbel values simultaneously. In particular, those coefficients are proposedherein as in the following:

γ11 =

∫ H

0Pb,x(z)zdz

My

=My

My

= 1

γ12 =

∫ H

0Pb,y(z)zdz

M∗x

=Mx(te)

M∗x

γ13 =

∫ H

0Tb(z)dz

M∗z

=Mz(te)

M∗z

(7)

where te is the time instant so that My(te) = My. The coefficients just defined belong to thequasi-static moments and torque. By the aim of considering the resonant loads as well, it wouldbe necessary to compute the above coefficients by the resultant forces logged at the base ofaeroelastic models, so involving also the effect of the structure motion. Since, the tested modelswere rigid within the present experimental tests, the same coefficients are assumed for resonantmoments and torque. This assumption could appear further arbitrary. Nevertheless, this issue isstill under examination and it will be one of the matters of future researches. By this hypothesis,the total loads, at the critical angles of Mx, My and Mz, will be finally written as:

Pb(z)|α(My) = γ11Px,tot(z)i1 + γ12P∗y,tot(z)i2 + γ13T

∗tot(z)i3

Pb(z)|α(Mx) = γ21P∗x,tot(z)i1 + γ22Py,tot(z)i2 + γ23T

∗∗tot(z)i3

Pb(z)|α(Mz) = γ31P∗∗x,tot(z)i1 + γ32P

∗∗,tot(z)i2 + γ33Ttot(z)i3

(8)

The procedure just described refers to the mono-variate extreme theory, as the Gumbel valuesof each moments and torque have been computed independently. By this theory, one of thestochastic processes under examination crosses the respective threshold without regards of whatthresholds are crossed by the other processes. Going by this, we would say that each Gumbelvalue is independent to the others. The coefficients γmn for the Garibaldi-Repubblica tower arereported in the following matrix Γ, for each critical angle:

Γ =

γ11 γ12 γ13

γ21 γ22 γ23

γ31 γ32 γ33

=

1 1 .996.999 1 .994.997 1 1

(9)

11


In the case of the Garibaldi-Repubblica tower, it is possible to see by the matrix Γ that thedesign loads are practically equal to their respective Gumbel loads, for each critical angle.Nevertheless, since this results refers to the mono-variate extreme theory, they might actuallybe too conservative or, at least, unlikely. For this reason, it might be better to apply the multi-variate extreme theory, in order to have the most likely coefficients combinations.Assuming αcr = α(My), given M = [My(t),Mx(t),Mz(t)] = [M1,M2,M3], the multi-variateextreme value distribution of M is given by (see e.g. Ref.[6]):

P[

3⋂i=1

(Mmi≤ ξi)

]= exp [−(λ1 + λ2 + λ3 + λ12 + λ13 + λ23)] (10)

where Mmi= max(min)0<t<tiXi(t) (i = 1 . . . 3). In Eq.(10) the joint extreme value distribu-

tion can be also understood as the probability distributions of zero crossings of the thresholdsξi during a certain time T . The parameters λi are expressed as in the following:

λi =

∫ T

0

∫ +∞

−∞

∫ +∞

−∞miδ(mi − ξi)pMi,Mi

(mi, mi; t)dmidmidt (11)

λij =

∫ T

0

∫ T

0

∫ +∞

−∞

∫ +∞

−∞

∫ +∞

−∞

∫ +∞

−∞mimjδ(mi − ξi)δ(mj − ξj)×

× pMi,Mj ,Mi,Mj(mi,mj, mi, mj; t1, t2)dmidmjdmidmjdt1dt2 (12)

In the case in which M is a stationary vector, as it is herein, Eqs. (11, 12) become:

λi =

∫ T

0

dt

∫ +∞

−∞

∫ +∞

−∞miδ(mi − ξi)pMi,Mi

(mi, mi)dmidmi = νi(ξi)T (13)

λij =

∫ T

0

∫ +∞

−∞

∫ +∞

−∞

∫ +∞

−∞

∫ +∞

−∞mimjδ(mi − ξi)δ(mj − ξj)×

× pMi,Mj ,Mi,Mj(mi,mj, mi, mj; t2 − t1)dmidmjdmidmjd(t2 − t1) = νij(ξi, ξj)T (14)

where νi(ξi) is the expected number of threshold crossings per unit time for Mi, and νij(ξi, ξj)is the expected number of contemporaneous threshold crossings of Mi and Mj per unit time.Moreover, the probability of the first crossing of the thresholds (ξ1, ξ2, ξ3), by both positive ornegative slopes, during the time T is

PT (t) = 1− exp [−(ν1 + ν2 + ν3 + ν12 + ν13 + ν23)T ] = exp (−νT ) (15)

The probability density of T follows by differentiating of Eq.(15), so obtaining:

pT (t) = ν exp (−νT ) (16)

Equation (16) may be used to compute the statistical properties of the first-passage time T . Inparticular, the mean and the variance of T are given by the following formula, respectively:

E [T ] =

∫ +∞

0

tpT (t)dt =1

ν(17)

E[T 2

]=

∫ +∞

0

(t− 1

ν

)2

pT (t)dt =1

ν2(18)

12


(a) γ11, γ12, γ13 at α(My) = 157.5o (b) γ11, γ12, γ13 at α(My) = 157.5o

(c) γ21, γ22, γ23 at α(Mx) = 247.5o (d) γ21, γ22, γ23 at α(Mx) = 247.5o

(e) γ31, γ32, γ33 at α(Mz) = 315o (f) γ31, γ32, γ33 at α(Mz) = 315o

Figure 14: Isosurfaces of ν(ξ1, ξ2, ξ3) supplying the 50-years combination coefficients for theGaribaldi-Repubblica tower obtained by using the base resultants moments and torque withinpressure-map integrations (Left), and aerodynamic-balance tests (Right). u: γij by monovari-ate extreme theory

13


Finally, by fixing the return period E [T ] = T , it is possible to find all the thresholds setsoccurring once every T years.Hereafter, by the aim of using the same notation for the loads combinations, the coefficients γij

will be re-written as

γii =ξ∗iiMi

γij =ξ∗ijM∗

j

(19)

By considering M1 = My, for instance, ξ11, ξ12 and ξ13 constitute the set of moments and torqueoccurring once in T years at αcr = α(My), within the multi-variate extreme theory. In general,M1 is the design value of M1 = My, whereas M∗

2 and M∗3 are, respectively, the Gumbel values

of Mx and My at α(My), computed by the univariate extreme theory.Herein, the return period has been fixed to 50 years, T = 50 years, so all the sets of (ξ1, ξ2, ξ3),supplying ν = 0.02 according to Eq.(17), have been computed for the base resultant momentsand torque obtained by integrating the pressure fields and by means of aerodynamic-balancetests. In Fig.14, the isosurfaces of ν = ν(ξ1, ξ2, ξ3), supplying all the 50-years-thresholds setsfor a certain critical wind angle of attack, are presented. As it is shown in Fig.14, the coefficientssets computed by the univariate extreme theory (see. Eq.(9)) actually never happen and, aboveof all, they don’t appear as the safest combination for designing the tower under examination(the Garibaldi-Repubblica one).

6 CONCLUSIONS

Within the result obtained in the present paper, it is possible to highlight the importance ofperforming both pressure-tap measurements and aerodynamic-balance tests for defining equiva-lent static wind-induced loads acting on tall buildings. The comparison of the base result forceslogged by both methods can draw attention to the accuracy of studying wind-induced pressurefields by means of pressure-tap measurements. Those fields, in fact, depend strongly on thestructure aerodynamic and on the impact of the surrounding urban contest on it, as shown in theforegoing. Moreover, by the similitude criteria necessary to switch from model to full scale, itmight be possible that the frequency band of the stochastic pressures, logged by pressure-taptests, is not enough to cover a range of frequencies which itself would let the assessment ofthe resonant structure response at full scale. In the case, high-frequency-aerodynamic-balancetests become a must within this aim. In fact, the resonant loads obtained herein are sensitivelydifferent among each other.Moreover, the use of the mono-variate extreme theory for defining the wind-induced loads andtheir combinations might be unlikely and not the safest solution. The results of the multi-variateextreme theory show itself to be more efforts expensive, but absolutely the most likely and safestway for designing structure whose importance is as high as design responsibilities are.

7 ACKNOWLEDGMENTS

The dynamic characteristics of the Garibaldi-Repubblica tower (eigenmodes and eigenfre-quencies) have been taken by the degree thesis of Ing. F. Giovannelli and Ing. E. Tomberli whoare grateful acknowledged.

REFERENCES

[1] E.J. Gumbel. Statistic of extremes. Columbia University Press, New York, 1958.

14


[2] A. G. Davenport. Missing links in wind engineering. Proc., 10th ICWE, Copenhagen, 18,1999

[3] C. Drybre, S.O. Hansen. Wind loads on structures Wiley, New York, 1997

[4] Y. Zhou, A. Kareem. Gust loading factor: new model. Journal of Structural Engineering127:2, 168-175, 2001

[5] Y. Zhou, T. Kijewski, A. Kareem. Aerodynamic loads on tall buildings. Journal of Struc-tural Engineering 129:3, 394-404, 2003

[6] S. Gupta, C.S. Manohar. Multivariate estreme value distributions for random vibrationapplications. Journal of Engineering Mechanics. 131:7, 2005

15

Equivalent static wind loads on tall buildings.pdf

Documents

Transcript of Equivalent static wind loads on tall buildings.pdf