EQUIVALENT STATIC WIND LOADS ON TALL BUILDINGS

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1 BBAA VI International Colloquium on: Bluff Bodies Aerodynamics & Applications Milano, Italy, July, 24 8 EQUIVALENT STATIC WIND LOADS ON TALL BUILDINGS Stefano Pastò, Luca Facchini, Lorenzo Procino, Paolo Spinelli CRIACIV/Dipartimento di Ingegneria Civile e Ambientale Università di Firenze, via S. Marta 3, 5133 Firenze, Italy s: stefanop@dicea.unifi.it, luca.facchini@dicea.unifi.it, lorenzo.procino@pin.unifi.it, paolo.spinelli@dicea.unifi.it Keywords: Random Vibrations, Tall Buildings, Wind-Induced Loads, Wind-Exposed Structures; Wind Tunnel Abstract. The results of experimental campaigns on two tall buildings are presented. Tests have been carried out in the CRIACIV boundary layer wind-tunnel by pressure-tap measurements and aerodynamic-balance tests, on both models. Numerical studies have been performed, afterwards, to define the equivalent static background and resonant wind-induced loads considering the structure three-dimensionality. The multivariate extreme theory has been adopted to propose design wind loading combinations. 1 INTRODUCTION Among the tall buildings already built in Italy, those studied herein are of great concern within wind-exposed structural design as, most probably, they are going to be the tallest Italian buildings. So, a careful design against wind-induced actions is a must in these cases. In particular, the subjects of the present paper are the towers of UNIPOL (125m tall) located in Bologna, and Piazza Garibaldi-Repubblica (m tall) located in Milano. The towers have been tested in the CRIACIV boundary-layer wind-tunnel by means of pressure-tap measurements and aerodynamic-balance tests. 2 EXPERIMENTAL TESTS The typical mean wind profile of an urban zone (β =.2274 within the exponential profile) has been reproduced in wind tunnel, as well as the atmospheric turbulence, within the well-known approximations in simulating real turbulence length scales. Moreover, the towers models and the surrounding urban contexts have been reproduced and placed in the text-section of the wind-tunnel, as shown in Fig. 1. In particular, both the models are 1:35 scaled and equipped by 125 and pressure taps in the case of the Unipol and Garibaldi-Repubblica towers, respectively (see Fig. 2). Pressure fields have been logged within 16 wind directions, equally spaced by 22.5 degrees, by a sampling frequency of 25Hz during 3 seconds. 1

2 UNIPOL tower Figure 1: Showing of the wind-tunnel samples Garibaldi-Repubblica tower 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 coefficients have been computed. The extremes values (maximum and minimum values) belong to a return period of 5 years. In general, mean and extreme pressure coefficients should be agreed upon as design values. Some pressure maps for both towers are shown in Figs. 3, 4 and 5. 2

3 Mean C p Mean C p Figure 3: Mean pressure maps at wind angle of attacks α = o and α = 11 o for the Garibaldi- Repubblica tower (left) and the Unipol tower (right), respectively Gumbel Maximum C p Gumbel Maximum C p Figure 4: Maximum pressure maps at wind angle of attacks α = o and α = 11 o for the Garibaldi-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 be used as design pressure maps. In fact, the mean values alone let just the assessment of the static structural response, whereas the extreme maps, although they take into account the fluctuating component of the wind-induced pressures, might lead to a structural design too much conservative 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 extreme theory, 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 logged by several pressure taps, analyses based on the multivariate extreme theory might be incredibly onerous compared to other fields of application. Moreover each structural response should be computed for each set of pressure coefficients carried out by this theory, in order to obtain the worst situation. On the other hand, it is possible to use this theory reducing drastically the number of time histories being processed by referring to the base resultant moments and torque, as 3

4 it is being proposed later on. From the foregoing it follows that those extreme maps might not be reliable for designing windexposed structure. So, other procedures might be by suitable. For instance, by computing a certain 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 C p Gumbel Minimum C p Figure 5: Minimum pressure maps at wind angle of attacks α = o and α = 11 o for the Garibaldi-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 structural 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 at the structure bases. By considering the base resultant forces as the structural responses under examination, it has been possible to map the pressure fields at the time instant supplying the extreme design values of the respective forces. Each of those extreme values has been found independently, that is by neglecting common threshold crossings. The results are reported in Figs. 6. The above design maps should be agreed upon as quasi-static ones, as they derive from the integration of the pressure fields logged on rigid wind-tunnel models. In general, the equivalent static background wind loads can be obtained by distributing the base moments and torque to 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 equivalent static resonant wind-loads starting from the time-histories of the base resultant moments and torque. Since the structure motion is considered, the base resultants must show frequency contents in a frequency band involving the structure eigenfrequency at full scale. In order to guarantee 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 (see e.g. Ref. [3]). Nevertheless, despite the sampling frequency of the pressure maps was quite high (25Hz), it was not enough to make the base forces have frequency contents in the desired band. So, the base resultant forces were also logged by means of aerodynamic-balance tests. 4

5 Map for maximum F x at α = o Map for maximum F x at α = 11 o Map for maximum M y at α = o Map for maximum M y at α = 11 o Map for minimum F y at α = o Map for minimum F y at α = 67.5 o Map for maximum M x at α = o Map for minimum M x at α = 67.5 o Map for maximum M z at α = 315. o Map for minimum M z at α = 67.5 o Figure 6: Design pressure maps providing the design values of the base resultant force. Left) Garibaldi-Repubblica tower. Right) UNIPOL tower 5

6 Figure 7: Aerodynamic balance 3 AERODYNAMIC BALANCE TESTS The base resultant forces and moments (along-wind and across-wind forces, moments and torque) of both towers have been logged by means of an aerodynamic balance. The 5-components aerodynamic balance is shown in Fig. 7. Tests have been carried out by means of a sampling frequency of Hz (4 times the frequency used for logging the pressure maps). Comparisons between the base forces obtained by integrating the pressure fields, and by the aerodynamic balance, are reported in Figs. 8, 9 and 1. In those figures, forces and moments 2 x 15 1 x M z (m 3 ).5 M z (m 3 ) M z vs. α M z vs. α Figure 8: Base resultant pseudo-torques varying the wind angle of attack (α). Left) Garibaldi- Repubblica tower. Right) Unipol tower. Legend: : Aerodynamic-Balance tests; E: Pressurefield integrations should be understood as pseudo-forces and pseudo-moments as their measure units disagree with the real ones. To switch to the full scale forces and moments, it is just necessary to multiply 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 fields as the number of pressure taps is finite and each tap belongs to a certain influence area. On the contrary, base forces obtained by aerodynamic-balance tests are much more reliable as they belong to the continuous wind-induced pressure fields. So, differences may show themselves 6

7 in comparing the forces obtained within the two procedures. In particular, such differences are more marked for the Garibaldi-Repubblica tower than for the Unipol tower. This might highlight how relevant could be not only the aerodynamic of the structure itself, but also the influence of the urban context: in the case of the Garibaldi-Repubblica tower, the urban context is 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 loads taking into account structure resonances. The procedure is shown in the next section F x (m 2 ) 5 F x (m 2 ) F x vs. α F x vs. α 8 x 15 4 x M y (m 3 ) 4 3 M y (m 3 ) M y vs. α M y 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: : Aerodynamic-Balance tests; E: Pressure-field integrations 7

8 3 F y (m 2 ) F y (m 2 ) F y vs. α F y vs. α 6 x 15 3 x M x (m 3 ) M x (m 3 ) M x vs. α M x vs. α Figure 1: Base resultant pseudo- forces and moments varying the wind angle of attack (α). Left) Garibaldi-Repubblica tower. Right) Unipol tower. Legend: : Aerodynamic-Balance tests; 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 associated to a certain mode shape, can be computed by (see Ref. [5]) for bending moments and P r (z) = M r T r (z) = T r o o m(z)φ i (z) m(z)φ i (z)zdz I(z)θ j (z) I(z)θ j (z)dz for torque. In Eqs. (1, 2), m(z) and I(z) are the structure mass and mass moment of inertia for unit height, respectively; φ i (z) and θ j (z) are the generic eigenmodes associated to bending and torque deformations, respectively. The resonant moments, M R, and torque, T R, are estimated amplifying the quasi-static moments by the structure transfer function at the bending (1) (2) 8

9 eigenfrequency, f i, and torque eigenfrequency, f j, respectively: M r = g ri π 4ν i f i S M (f i ) T r = g rj π 4ν j f j S T (f j ) (3) where ν k (k = i, j) is the damping ratio associated to the considered mode shape; S M and S T are the quasi-static spectra of base moment and torque, respectively; g r = 2 ln f k t / 2 ln f k t (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, moments and torque might show frequency contents at the structure eigenfrequencies, so contributing to amplify the structure response. In this case, M R and T R, in Eq.(3), must be evaluated at the same frequency (e.g. f i for the mode φ i (z)) and then projected on the respective components of the i th mode shape under consideration. Nevertheless, if the eigenmodes are uncoupled, that is they just show deformations along a certain direction (x, y or z), the definition of the equivalent static wind loads in Eq.(2) only applies to the relevant moment or torque which yields deformation along the eigenmode direction. In the latter case, the equivalent static wind-induced loads, including quasi-static (P b,x (z), P b,y (z) and T b,z (z)) and resonant components (P r,x (z), P r,y (z) and T r,z (z)), may be written as following: π P x,tot (z) = P b,x (z) + P r,x (z) = P b,x (z) + g ri f i S My (f i ) 4ν i π P y,tot (z) = P b,y (z) + P r,y (z) = P b,y (z) + g rj f j S Mx (f j ) 4ν j π T tot (z) = T b (z) + T r (z) = T b (z) + g rk f k F S Mz (f k ) 4ν k o o o m(z)φ i (z) m(z)φ i (z)zdz m(z)φ j (z) m(z)φ j (z)zdz m(z)φ k (z) m(z)φ k (z)zdz where φ i (z), φ j (z) and φ k (z) are the mode shapes related to the eigenfrequencies f i, f j and f k, mainly developing along x, y and around z, respectively. The quasi-static loads are obtained by integrating the wind-induced pressure fields. In particular, the following procedure have been adopted. For instance, in order to obtain ˆP b,x (z) we are interested to find the pressure map corresponding to quasi-static design (Gumbel) value of the base resultant ˆMy at the wind angle of attack α cr = α( ˆM y ). So, the time instant (t e ), at which the time-history M y (t) crosses the Gumbel threshold (by positive or negative slopes according to the sign of the design value) is searched. Then, the pressure map belonging to t e is integrated along x at each floor, assuming full scale structure by rigid slabs. Since structures are 3D, we can also obtain, by the pressure map under examination, P b,y (z) and T b,z (z), related to M x and M z at α cr = α( ˆM y ), respectively. Clearly, in general, P b,y (z) ˆP b,y (z) and T b (z) ˆT b (z). The respective resonant components of those loads have been computed by the procedure described above, so getting the total wind-induced loads according to Eq.(4) in the case of ˆM y. The procedure has been repeated for ˆM x and ˆM z. In this way, the loads for each α cr (at which moments and torque approach their design values; see Fig. 6) have been found. The results are reported in Figs. 11, 12 and 13 for the Garibaldi-Repubblica tower. In particular, the resonant forces and torques have been computed by means of the quasi-static base resultants obtained by both pressure-map integrations and aerodynamic-balance tests. Those loads are further different. This is ascribed to different frequency contents of moments and torque spectra, within the two procedures. 9 (4)

10 5 15 P x,b.5 1 P x,r 5 15 P x,tot P y,b.5 1 P y,r P y,tot T b (z) [knm/m] 15 5 T r (z) [knm/m] T tot (z) [knm/m] (a) P x,b (z), P x,r (z) and P x,tot (z) (b) P y,b (z), P y,r (z) and P y,tot (z) (c) T b (z), T r (z) and T tot (z) Figure 11: Garibaldi-Republica tower: equivalent static wind loads at α = o (M x = ˆM x ). Legend: resonant loads obtained by aerodynamic-balance time histories 3 5 P x,b.5 1 P x,r P x,tot 5 P y,b 1.5 P y,r 5 P y,tot 5 15 T b (z) [knm/m] 5 T r (z) [knm/m] 5 15 T tot (z) [knm/m] (a) P x,b (z), P x,r(z) and P x,tot (z) (b) P y,b (z), P y,r(z) and P y,tot (z) (c) T b (z), T r(z) and T tot (z) Figure 12: Garibaldi-Republica tower: equivalent static wind loads at α = o (M y = ˆM y ). Legend: resonant loads obtained by aerodynamic-balance time histories P x,b.5 1 P x,r P x,tot P y,b.5 1 P y,r P y,tot 15 5 T (z) [knm/m] b 15 5 T (z) [knm/m] r 15 5 T (z) [knm/m] tot (a) P x,b (z), P x,r(z) and P x,tot (z) (b) P y,b (z), P y,r(z) and P y,tot (z) (c) T b (z), T r(z) and T tot (z) Figure 13: Garibaldi-Republica tower: equivalent static wind loads at α = 315 o (M z = ˆM z ). Legend: resonant loads obtained by aerodynamic-balance time histories 1

11 5 STATISTICAL COMBINATION OF DESIGN WIND-INDUCED LOADS For the sake of clarity, let s keep considering ˆM y. The quasi-static 3D-loading, at α cr = α( ˆM y ), may be written as follows: P b (z) α( ˆMy) = ˆP b,x (z)i 1 + P b,y (z)i 2 + T b (z)i 3 (5) where i 1, i 2 and i 3 are the versors along the Cartesian coordinates x, y and z. The total load in Eq.(5) may be still re-written by introducing the following notation: P b (z) α( ˆMy ) = γ 11 ˆP b,x (z)i 1 + γ 12 P b,y(z)i 2 + γ 13 T b (z)i 3 (6) where Pb,y and T b are the loads reproducing to the Gumbel values of M x(t) and M z (t) at α( ˆM y ), that is Mx and Mz So, in order to get the latter loads, it is necessary to integrate the maps at the instant during which M x (t) = Mx and M z (t) = Mz at α( ˆM y ). The coefficients γ 11, γ 12 and γ 13 have to be understood as coefficients to combine ˆP b,x (z), Pb,y (z) and T b (z) statistically, as M x (t), M y (t) and M z (t) are not fully correlated among each other and hence, generally, they do not approach their Gumbel values simultaneously. In particular, those coefficients are proposed herein as in the following: γ 11 = γ 12 = γ 13 = ˆP b,x (z)zdz ˆM y = ˆM y ˆM y = 1 P b,y(z)zdz M x T b(z)dz M z = M x(t e ) M x = M z(t e ) M z where t e is the time instant so that M y (t e ) = ˆM y. The coefficients just defined belong to the quasi-static moments and torque. By the aim of considering the resonant loads as well, it would be necessary to compute the above coefficients by the resultant forces logged at the base of aeroelastic models, so involving also the effect of the structure motion. Since, the tested models were rigid within the present experimental tests, the same coefficients are assumed for resonant moments and torque. This assumption could appear further arbitrary. Nevertheless, this issue is still under examination and it will be one of the matters of future researches. By this hypothesis, the total loads, at the critical angles of M x, M y and M z, will be finally written as: P b (z) α( ˆMy) = γ 11 ˆP x,tot (z)i 1 + γ 12 P y,tot(z)i 2 + γ 13 T tot(z)i 3 P b (z) α( ˆMx ) = γ 21P x,tot(z)i 1 + γ 22 ˆPy,tot (z)i 2 + γ 23 T tot(z)i 3 P b (z) α( ˆMz) = γ 31P x,tot(z)i 1 + γ 32 P (7),tot(z)i 2 + γ 33 ˆTtot (z)i 3 (8) The procedure just described refers to the mono-variate extreme theory, as the Gumbel values of each moments and torque have been computed independently. By this theory, one of the stochastic processes under examination crosses the respective threshold without regards of what thresholds are crossed by the other processes. Going by this, we would say that each Gumbel value is independent to the others. The coefficients γ mn for the Garibaldi-Repubblica tower are reported in the following matrix Γ, for each critical angle: γ 11 γ 12 γ 13 Γ = γ 21 γ 22 γ 23 = γ 31 γ 32 γ (9)

12 In the case of the Garibaldi-Repubblica tower, it is possible to see by the matrix Γ that the design 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 actually be too conservative or, at least, unlikely. For this reason, it might be better to apply the multivariate extreme theory, in order to have the most likely coefficients combinations. Assuming α cr = α( ˆM y ), given M = [M y (t), M x (t), M z (t)] = [M 1, M 2, M 3 ], the multi-variate extreme value distribution of M is given by (see e.g. Ref.[6]): [ 3 ] P (M mi ξ i ) = exp [ (λ 1 + λ 2 + λ 3 + λ 12 + λ 13 + λ 23 )] (1) i=1 where M mi = max(min) <t<ti X i (t) (i = ). In Eq.(1) the joint extreme value distribution can be also understood as the probability distributions of zero crossings of the thresholds ξ i during a certain time T. The parameters λ i are expressed as in the following: λ ij = λ i = T + + T T ṁ i δ(m i ξ i )p Mi,Ṁi (m i, ṁ i ; t)dm i dṁ i dt (11) ṁ i ṁ j δ(m i ξ i )δ(m j ξ j ) p Mi,M j,ṁi,ṁj (m i, m j, ṁ i, ṁ j ; t 1, t 2 )dm i dm j dṁ i dṁ j dt 1 dt 2 (12) In the case in which M is a stationary vector, as it is herein, Eqs. (11, 12) become: λ ij = λ i = T dt + + T ṁ i δ(m i ξ i )p Mi,Ṁi (m i, ṁ i )dm i dṁ i = ν i (ξ i )T (13) ṁ i ṁ j δ(m i ξ i )δ(m j ξ j ) p Mi,M j,ṁi,ṁj (m i, m j, ṁ i, ṁ j ; t 2 t 1 )dm i dm j dṁ i dṁ j d(t 2 t 1 ) = ν ij (ξ i, ξ j )T (14) where ν i (ξ i ) is the expected number of threshold crossings per unit time for M i, and ν ij (ξ i, ξ j ) is the expected number of contemporaneous threshold crossings of M i and M j per unit time. Moreover, the probability of the first crossing of the thresholds (ξ 1, ξ 2, ξ 3 ), by both positive or negative slopes, during the time T is P T (t) = 1 exp [ (ν 1 + ν 2 + ν 3 + ν 12 + ν 13 + ν 23 )T ] = exp ( νt ) (15) The probability density of T follows by differentiating of Eq.(15), so obtaining: p T (t) = ν exp ( νt ) (16) Equation (16) may be used to compute the statistical properties of the first-passage time T. In particular, the mean and the variance of T are given by the following formula, respectively: E [T ] = E [ T 2] = + + tp T (t)dt = 1 (17) ν ( t 1 ) 2 p T (t)dt = 1 (18) ν ν 2 12

13 (a) γ 11, γ 12, γ 13 at α( ˆM y) = o (b) γ 11, γ 12, γ 13 at α( ˆM y) = o (c) γ 21, γ 22, γ 23 at α( ˆM x) = o (d) γ 21, γ 22, γ 23 at α( ˆM x) = o (e) γ 31, γ 32, γ 33 at α( ˆM z) = 315 o (f) γ 31, γ 32, γ 33 at α( ˆM z) = 315 o Figure 14: Isosurfaces of ν(ξ 1, ξ 2, ξ 3 ) supplying the 5-years combination coefficients for the Garibaldi-Repubblica tower obtained by using the base resultants moments and torque within pressure-map integrations (Left), and aerodynamic-balance tests (Right). u: γ ij by monovariate extreme theory 13

14 Finally, by fixing the return period E [T ] = ˆT, it is possible to find all the thresholds sets occurring once every ˆT years. Hereafter, by the aim of using the same notation for the loads combinations, the coefficients γ ij will be re-written as γ ii = ξ ii ˆM i γ ij = ξ ij M j By considering ˆM 1 = ˆM y, for instance, ξ 11, ξ 12 and ξ 13 constitute the set of moments and torque occurring once in ˆT years at α cr = α( ˆM y ), within the multi-variate extreme theory. In general, ˆM 1 is the design value of M 1 = M y, whereas M 2 and M 3 are, respectively, the Gumbel values of M x and M y at α( ˆM y ), computed by the univariate extreme theory. Herein, the return period has been fixed to 5 years, ˆT = 5 years, so all the sets of (ξ 1, ξ 2, ξ 3 ), supplying ν =.2 according to Eq.(17), have been computed for the base resultant moments and torque obtained by integrating the pressure fields and by means of aerodynamic-balance tests. In Fig.14, the isosurfaces of ν = ν(ξ 1, ξ 2, ξ 3 ), supplying all the 5-years-thresholds sets for a certain critical wind angle of attack, are presented. As it is shown in Fig.14, the coefficients sets computed by the univariate extreme theory (see. Eq.(9)) actually never happen and, above of 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 of performing both pressure-tap measurements and aerodynamic-balance tests for defining equivalent static wind-induced loads acting on tall buildings. The comparison of the base result forces logged by both methods can draw attention to the accuracy of studying wind-induced pressure fields by means of pressure-tap measurements. Those fields, in fact, depend strongly on the structure aerodynamic and on the impact of the surrounding urban contest on it, as shown in the foregoing. Moreover, by the similitude criteria necessary to switch from model to full scale, it might be possible that the frequency band of the stochastic pressures, logged by pressure-tap tests, is not enough to cover a range of frequencies which itself would let the assessment of the resonant structure response at full scale. In the case, high-frequency-aerodynamic-balance tests become a must within this aim. In fact, the resonant loads obtained herein are sensitively different among each other. Moreover, the use of the mono-variate extreme theory for defining the wind-induced loads and their combinations might be unlikely and not the safest solution. The results of the multi-variate extreme theory show itself to be more efforts expensive, but absolutely the most likely and safest way for designing structure whose importance is as high as design responsibilities are. 7 ACKNOWLEDGMENTS The dynamic characteristics of the Garibaldi-Repubblica tower (eigenmodes and eigenfrequencies) have been taken by the degree thesis of Ing. F. Giovannelli and Ing. E. Tomberli who are grateful acknowledged. (19) REFERENCES [1] E.J. Gumbel. Statistic of extremes. Columbia University Press, New York,

15 [2] A. G. Davenport. Missing links in wind engineering. Proc., 1th 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 Engineering 127:2, , 1 [5] Y. Zhou, T. Kijewski, A. Kareem. Aerodynamic loads on tall buildings. Journal of Structural Engineering 129:3, 394-4, 3 [6] S. Gupta, C.S. Manohar. Multivariate estreme value distributions for random vibration applications. Journal of Engineering Mechanics. 131:7, 5 15