Generalized gust-front factor: A computational framework for wind load effects

Engineering Structures 48 (2013) 635–644

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Generalized gust-front factor: A computational framework for wind load effects Dae Kun Kwon ⇑, Ahsan Kareem NatHaz Modeling Laboratory, 156 Fitzpatrick Hall, Department of Civil & Environmental Engineering & Earth Sciences, University of Notre Dame, Notre Dame, IN 46556, USA

a r t i c l e

i n f o

Article history: Received 16 July 2012 Revised 2 December 2012 Accepted 20 December 2012 Available online 26 January 2013 Keywords: Wind loads Nonstationary Gust-front Gust-front factor Thunderstorm Downburst Codes and standards

a b s t r a c t This paper presents a unified framework of a generalized gust-front factor for modeling winds in gustfronts and their attendant load effects on structures. This is analogous to the gust loading factor, a widely recognized format used world-wide in codes and standards, for the treatment of gusts in conventional boundary layer winds. The generalized gust-front factor encapsulates dynamic features inherent in wind load effects in gust-fronts originating from winds in downbursts/thunderstorms. In view of the computational complexity of the formulation and the need to promote the usage of such a framework, a webbased portal is provided. The portal at http://gff.ce.nd.edu can be conveniently utilized to evaluate the generalized gust-front factor in conjunction with international standards. The proposed generalized gust-front factor reduces to conventional gust loading factor in the case of boundary layer winds, thus, it represents a most generalized dynamic wind load effects modeling framework on structures that is compatible with the format used in ASCE 7 or any other international standard. Ó 2012 Elsevier Ltd. All rights reserved.

1. Introduction The mechanics of gusts associated with convective gust-fronts differs significantly from traditional turbulence in boundary layer winds both in its kinematics and dynamics. The key distinguishing attributes are the contrasting velocity profile with height, a rapid increase in speed and the statistical features of the energetic gusts in the wind field. In gust-fronts, the traditional velocity profile does not exist; rather it bears an inverted velocity profile with its maxima near the ground potentially exposing low- to mid-rise structures to higher wind loads [1–3]. Furthermore, such a change in the approach flow profile/kinematics, even in a steady state flow, would introduce a major change in the flow-structure interaction that may differ significantly from the corresponding boundary layer flow case. This is compounded by the inherent transient nature of energetic convective gusts that rapidly increase in amplitude and direction, raising serious questions regarding the applicability of conventional aerodynamic loading theories. The nonstationarity is the critical issue in these wind events, which has been examined utilizing full-scale measurements (e.g., [4,5]). To account for the buffeting effects of gustiness in turbulent boundary layer winds for structural loading, most international codes and standards have adopted the concept of a gust loading factor which was first introduced by Davenport [6] and has been ⇑ Corresponding author. Tel.: +1 574 631 5380; fax: +1 574 631 9236. E-mail addresses: [email protected] (D.K. Kwon), [email protected] (A. Kareem). 0141-0296/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.engstruct.2012.12.024

extensively improvised by a host of researchers (e.g., [7]) and more recently recast into a new format by Zhou and Kareem [8]. In comparison with the boundary layer winds that have generally been regarded as stationary, gust-front winds originating from a downburst/thunderstorm exhibit distinct nonstationarity, i.e., rapid changes in wind speed during a short time interval which may be accompanied by changes in direction. The significance of these transient wind events, as they relate to subsequent load effects, can be readily surmised based on the analysis of thunderstorm databases both in the U.S. and around the world, which suggest that these winds actually represent the design wind speed for many locations. In order to realistically capture characteristics of gust-front winds and their attendant load effects, Kwon and Kareem [1,2] proposed a new analysis framework, gust-front factor, which expresses a description of the genesis of the overall wind load effects on structures under both gust-front and boundary layer winds. It has been designed to be used in conjunction with the existing codes and standards, especially ASCE 7 standard [9]. In the case of conventional boundary layer winds it simply reduces to the gust loading factor. In this study, a generalized gust-front factor framework is introduced, which is intended not only to analytically encapsulate dynamic load effects associated with gust-front winds independent of any reference design standard, e.g., ASCE 7, but also to highlight other general features like those found in conventional gust loading factor scheme [6]. Accordingly, the generalized gust-front

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factor seamlessly represents wind effects of both conventional boundary layer and gust-front winds in a consistent and unified manner. A web-based portal for the gust-front factor and associated loads available at http://gff.ce.nd.edu can be utilized to evaluate the generalized gust-front factor. 2. Modeling of gust-front winds 2.1. A time function and a vertical profile of gust-front winds In contrast with the boundary layer winds, the gust-front winds are nonstationary due to their transient characteristics. Therefore, the stationary wind model typically used in boundary layer winds may not be valid for gust-front winds which may be described in terms of time-varying parameters. Accordingly, the nonstationary wind model has been introduced and given below that includes storm-translation speed (Vs-m), which it may be assumed to be constant during a gust-front wind event and its spatial distribution with height is uniform as well (e.g., [1–3]): In this study, analytical/empirical models of downburst winds that characterize their spatiotemporal features are employed in terms of a time function and a vertical profile of gust-front winds [1–3]. Rather than relying on the time-varying features of only a few storms records to represent the transient nature of the storm (e.g., [1,2,10–12]), for generality, this study employs a half-sine wave to describe this feature. Even though it may not represent the exact time variation of winds in a typical gust-front, it captures the underlying feature potentially responsible for enhanced loads. At a future time, this description may be revised once a sufficient number of measurements become available and an acceptable description of this function is arrived at based on an ensemble average of such observations. The function is defined as:

V G-F ðtÞ ¼ sin



p

td

t

 ð1Þ

where, td = pulse duration of the excitation. The simplicity of expression is an attractive feature as it requires only a single parameter, td, to define the time function in a gust-front wind, while encapsulating the essential features of a sudden rise and drop in wind speed. The vertical profile of gust-front winds is critical in evaluating the wind effects on structures, however, very limited full-scale data along the height is available to identify and reliably establish a description of the vertical profile. From a practical viewpoint, the worst scenario for wind loads on structures may be the highest wind speed which occurs at about one downdraft jet diameter from its point of impact [12]. Note that several analytical models have been proposed for vertical profile of gust-front winds (e.g., [13–17]). Although there are some discrepancies among models, especially a profile over zmax (Eq. (2)), all models moderately fit well with a limited full-scale data. Thus, without loss of generality, the following model proposed in Vicroy [15] is utilized in this study due to its simplicity:

V GF ðzÞ ¼ 1:354  V max ½e0:22ðz=zmax Þ  e2:75ðz=zmax Þ 

ð2Þ

where Vmax = maximum horizontal wind speed, zmax = a height where Vmax occurs.

[VB-L,3-s(z)], e.g., ASCE 7 standard. Since zmax and Vmax in the Vicroy model (Eq. (2)) are unknown, it is necessary to establish a criterion to relate the velocity profile in a gust-front wind and a boundary layer for design considerations. In this study, two criteria are considered: (i) gust-front wind speed at 10 m height, VG-F(10), is set equal to the boundary layer gust speed at 10 m [Criterion 1: VG-F(10) = VB-L,3-s(10)]; (ii) the maximum gust-front wind speed (Vmax) is equal to gust speed at the gradient height (zG) in boundary layer winds [Criterion 2: Vmax = VB-L,3-s(zG)]. One may conveniently introduce additional criterion that may better reflect the data or may meet other site-specific requirements when it becomes available in the future. Although the vertical profile model (Eq. (2)) is, in reality, an analytical/empirical model based on limited full-scale data (JAWS), it represents open terrain exposure [14,15], e.g., exposure C in ASCE 7. Thus, it is expected that there is a certain terrain roughness effect on both zmax and Vmax in the Vicroy model (Eq. (2)), i.e., these two parameters may change with exposure categories in the codes & standards, whereas the model constants are assumed to be constant irrespective of the terrain roughness. Even though this terrain roughness effect on zmax and Vmax has been observed by some researchers through wind-tunnel experiments, the results have been limited to highlighting trends in different terrain roughnesses without any quantitative estimates. These general trends exhibited higher zmax (e.g., [18]) and lower Vmax (e.g., [17,19]) in a built-up terrain. With proper assumptions, vertical profile of gust-front winds for an arbitrary terrain exposure condition is established; details omitted here for the sake of brevity can be found in Kwon and Kareem [1,2]. Table 1 provides, for example, Vmax and zmax for different terrains, assuming V3-s to be 40 m/s based on ASCE 7 [9]. These results follow the trends noted in experimental observations, i.e., as terrain roughness increases, Vmax decreases but zmax increases. Similar tables can be established for other international codes and standards. 2.3. Discussion on the maximum horizontal wind speed, Vmax In the design wind speed, mean recurrence interval (MRI) or return period is an important issue: typically 50-year wind speeds, i.e., annual probability of 0.02, are used in boundary layer winds for survivability design. However, currently there is no definitive consensus for MRI in the gust-front winds due to a lack of full-scale data. Fujita [10] reported average probability of peak wind speeds in downbursts based on full-scale data at 10 m height in measurement programs NIMROD (1978) and JAWS (1982) involving 27 portable automated mesonet (PAM) stations. He suggested expressions for the probability of peak wind speeds:

Log 10 P ¼ 0:620  0:0873W

: NIMROD

Log 10 P ¼ 0:216  0:0902W

: JAWS

where P = probability of occurrence (per year); W = peak wind speed (m/s). Assuming that this relation is valid for other similar exposure condition (e.g., exposure C in ASCE 7), for comparison purpose, let P = 0.02/year, which corresponds to 50 year MRI, then peak wind speed at 10 m height in the case of NIMROD yields 26.56 m/s.

Table 1 Vmax and zmax for terrain exposures (V3-s is assumed as 40 m/s). Vmax (m/s)

2.2. Criteria for profile comparison in gust-front and boundary layer winds The vertical profile model of a downburst describes a short time averaged maximum mean wind speed at a height, which may be treated as a gust profile as used in the boundary layer wind case

ð3Þ

Exposure Exposure Exposure Exposure

A B C D

zmax (m)

Criterion 1

Criterion 2

71.26 81.29 89.47 93.06

45.15 51.50 56.68 58.96

100.58 80.47 60.35 46.94

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According to Eq. (2) with VG-F(10) = 26.56 m/s and zmax = 60.35 m (Table 1), Vmax is estimated as 59.41 m/s, which is very close to the value of Criterion 2 at exposure C, 56.68 m/s (Table 1). Thus, Criterion 2 proposed in Kwon and Kareem [1,2] may be reasonable estimate in the context of 50 year MRI. Examining from a different perspective, most codes and standards offer a wind map generally based on 50 year MRI, e.g., wind map in ASCE 7 is based on 3-s gust speed (i.e., short time average) for exposure C. It is quite likely that there may be many locations where small-scale but intense storms such as downburst/thunderstorm winds may have dominated the extreme wind records (e.g., [20,21]). In addition, recent full-scale measurements of downbursts and thunderstorm outflow field experiment [11] have shown that the peak wind speed at 10 m height tower was about 41 m/s, which is similar to a representative 3-s gust (40 m/s) for non-hurricane region in ASCE 7 [9]. In this regard, Criterion 1 [VG-F(10) = VB-L,3-s(10)] may also be acceptable assumption, though Criterion 1 shows much higher wind speed than Criterion 2 as shown in Table 1. Finally, it is noteworthy that in the past, a historic downburst event at Andrew Air Force Base in 1983, the peak wind speed of 67 m/s at 4.9 m height was observed. Based on this record, corresponding Vmax using Vicroy model is far above Criterion 1. Fujita [10] estimated that the chance of experiencing such a microburst is rare and four per year are expected within the contiguous US, yet this event may not be ignored due to its potential impact on structures. The choice of Criterion or Vmax in design procedure of gust-front winds will affect design loads significantly, however, as observed above a general consensus on the quantitative description involved is still evolving. Once such consensus is reached, it can be conveniently embedded in this framework.

respectively; gB-L = peak factor; rB-L(z) = root-mean-square (RMS) displacement by fluctuating wind components; GGLF = gust loading factor. As such, the gust loading factor accounts for the dynamics of wind fluctuations and any load amplification introduced by the dynamics of the structure, and this factor is utilized to estimate design loads of structure, i.e., ESWL. In the case of gust-front winds, structural displacement by gustfront winds [xG-F(z,t)] may be described in terms of nonstationary wind model that includes storm-translation effect [1–3]:

xG-F ðz; tÞ ¼ xG-F ðz; tÞ þ ~xG-F ðz; tÞ þ xs-m

ð5Þ

where, subscript G-F represents gust-front; superscripts – and  represent time-varying mean and nonstationary fluctuating components of displacement, respectively; xs-m = displacement introduced by the storm-translation speed. Accordingly, the corresponding maximum response can be approximately expressed as [1–3]:

max½xG-F ðz;tÞ  max½xG-F ðz;tÞ þ max½~xG-F ðz;tÞ þ xs-m   g  max½rG-F ðtÞ xs-m  max½xG-F ðz;tÞ 1 þ G-F þ max½xG-F ðz;tÞ max½xG-F ðz;tÞ ð6Þ where, gG-F and rG-F(t) represent mean peak factor [22] and timedependent RMS displacement by nonstationary fluctuating components of gust-front winds. In terms of the factorized analytical/empirical models of downburst winds [1–3], the maximum displacement of time-varying mean component may be divided into static and dynamic parts:

max½xG-F ðz; tÞ ¼ xst;G-F ðzÞ  max½xG-F ðtÞ

ð7Þ

Here, the generalized gust-front factor, denoted by GG,G-F, is defined as:

max½xG-F ðz; tÞ xst;G-F ðzÞ   max½xG-F ðtÞ g  max½rG-F ðtÞ xs-m ¼ 1 þ G-F þ xst;G-F ðzÞ max½xG-F ðz; tÞ max½xG-F ðz; tÞ

GG;G-F ¼ 3. Generalized gust-front factor With the exemplary acceptance of the gust loading factor [6] for capturing the dynamic wind effects introduced by buffeting action of wind and its popularity in design standards and codes worldwide, the framework for ‘‘Gust-Front Factor’’ (GG-F) [1,2] was proposed. It was intended to work in conjunction with the gust loading/effect formulation to encapsulate critical dynamic features of downburst winds. The current gust-front factor is represented by a product of four underlying factors to mimic the kinematic and dynamic features in a gust-front, however it is couched in ASCE 7 format [1–3]. In this format, the GG-F (Eq. (9a)) takes into account the following features: variation in the vertical profile of wind speed - kinematic effects factor (mean load effects), I1; dynamic effects introduced by the sudden rise in wind speed - pulse dynamics factor (rise-time effects), I2; nonstationarity of turbulence in gust-front winds - structural dynamics factor (nonstationary turbulence effects), I3; transient aerodynamics - potential load modification factor (transient aerodynamics effects), I4 [1–3]. For application to any code/standard, a generalized gust-front factor (GG,G-F) is introduced here that is independent of any reference code/standard format. Assuming that xB-L(z,t) is the structural displacement in boundary layer winds, the expected maximum response, max[xB-L(z,t)], can be expressed in terms of the gust loading factor format given in Davenport [6]:

max½xB-L ðz; tÞ ¼ max½xB-L ðzÞ þ ~xB-L ðz; tÞ   g  rB-L ðzÞ ¼ xB-L ðzÞ 1 þ B-L ¼ xB-L ðzÞ  GGLF xB-L ðzÞ

ð4Þ

where subscript B-L represents boundary layer; superscripts – and  represent mean and fluctuating components of displacement,

ð8Þ

In this format, GG,G-F can best capture the dynamic effects of gust-front winds akin to the concept of conventional gust loading factor. Note that strictly speaking, xs-m may be categorized as static component, but since its contribution becomes relatively small due to a large value in the denominator, i.e., the maximum displacement of time-varying mean component, this term may be treated small in the consideration of the overall static effect. For reference, the relationship between gust-front factor (GG-F) with associated underlying factors [1–3] and generalized gust-front factor (GG,G-F) can be described as:

GG-F ¼

st;G-F ðzÞ max½y G-F ðz; tÞ ½1 þ y   B-L ðzÞ st;G-F ðzÞ y y

g G-F max½rG-F ðtÞ G-F ðz;tÞ max½y

þ max½yys-m  G-F ðz;tÞ

GGLF C D;G-F  ¼ I 1  I 2  I3  I4 ð9aÞ CD   max½xG-F ðtÞ g  max½rG-F ðtÞ xs-m GG;G-F ¼ 1 þ G-F þ xst;G-F ðzÞ max½xG-F ðz; tÞ max½xG-F ðz; tÞ ¼ I2  ðI3  GGLF Þ

ð9bÞ

The (I3  GGLF) in GG,G-F indeed relates to nonstationary fluctuating component. Please note that in order to delineate the influence of CD,G-F on the GG-F, the drag coefficient in the gust-front and boundary layer winds formulations is separated by redefining response x as y in Eq. (9a), where y response is evaluated by setting drag force coefficient equal to unity. Regardless of this format, the relationship in Eq. (9b) remains valid, because in the formulation of GG,G-F (e.g., Eqs. (8), (9b)) the effect of drag coefficient (CD,G-F) cancels out in each term. It is worth noting that in the case of boundary layer winds, the sub-factors I2 and I3 become unity in the absence of

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Fig. 1. Schematic diagram of generalized gust-front factor framework.

gust-front wind effects. Accordingly, the resulting generalized gustfront factor (GG,G-F) reduces to conventional gust loading factor (GGLF) due to the relationship between the two factors shown in Eq. (9b). Finally, the design loads (ESWL) associated with the generalized gust-front factor are expressed in conventional format as used in the case of boundary layer winds:

F G;G-F ðzÞ ¼ FðzÞ  GG;G-F ¼

1 q  A  V 2G-F ðzÞ  GG;G-F  C D;G-F 2

ð10Þ

where q = air density; A = tributary area; CD,G-F = mean drag coefficient in gust-front winds (=I4  CD based on Eq. (9)). A schematic diagram that portrays the genesis of design wind loads in gust-front winds in terms of a generalized gust-front factor framework is given in Fig. 1. It is worth noting that the design loads defined in the gustfront factor and generalized gust-front factor yield the same result for ASCE 7 standard, if CD,G-F in Eq. (10) is treated in ASCE 7 format, i.e., windward and leeward pressure coefficients [9]. 4. Computational features

For cases in which the pulse duration (td) in the time function (Eq. (1)) is long enough, time-dependent frequency response function [M1(n,t), TDFRF], which is used to evaluate nonstationary RMS response, rG-F(t) in Eqs. (6), (8) and (9), may be decomposed into a product of an amplitude modulated function and conventional frequency response function [H1(n), TIFRF] (e.g., [23]). Under this specific condition of long duration, time-dependent mean-square response of fluctuating gust-front winds may be approximated as a combination of the background and resonant components, similar to those in boundary layer winds (e.g., [6]):

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi Z 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sf ;G-F ðn; tÞjM1 ðn; tÞj2 dn  F G-F ðtÞ BG-F þ RG-F

1 F G-F ðtÞ ¼ ðqBH  c1 Þ  ðCAÞ  ð1:354V max Þ2  V 2G-F ðtÞ k1 Z 1Z 1  0:22H=zmax Z 2 1 e  e2:75H=zmax Z1 ðCAÞ2 ¼ 0



ð11Þ

0

where Sf,G-F(n,t) = time-dependent wind force spectrum that may be treated as an evolutionary spectrum (e.g., [24]); M1(n,t) = timedependent frequency response function (TDFRF) (e.g., [1,2,23]);

ð12Þ

0

2 e0:22H=zmax Z1  e2:75H=zmax Z1 Z 1 Z 2 dZ 1 dZ 2

If the system is lightly damped, the frequency response function H1(n) peaks steeply at resonance, and the system acts like a narrow-band filter. Since the spectral density of the buffeting action of wind may be treated as a broad-band process, the resonant component of mean-square response can be approximated by (e.g., [6]):

RG-F ¼

4.1. Closed-form of nonstationary RMS response for long pulse duration

rG-F ðtÞ ¼

BG-F, RG-F = background and resonant components, respectively; k1 = modal stiffness; B, H = building width and height, respectively; c1 = a coefficient of amplitude modulating function; (CA) = a factor for normalized vertical joint acceptance function;

p 4f1

 n0  Su;s ðn0 Þ 

1 1  n0 B n0 H 1 þ 0:5  C x Vð10Þ 1 þ c3  C z Vð10Þ

ð13Þ

where f1 = damping ratio; n0 = natural frequency of structure; Su,s(n) = normalized Davenport wind spectrum; Cx, Cz = exponential decay coefficients; c3 = a height-dependent polynomial approximation of vertical joint acceptance function which can be expressed as a function of a building height. It is known that exponential decay coefficients Cx and Cz in V(10)-based coherence functions are height-dependent coefficients, however, an averaged value of the coefficients equal to 6 is taken. This is approximated based on typical values from height of exponential coefficients (e.g., [25]). This will reflect correlation with height in gust-front winds higher than typical boundary layer winds, which is premised on the nature of the mechanism underlying each wind event. The background component of the mean-square response in boundary layer winds could be found from the load spectrum, which does not include the resonant peak and joint acceptance functions are treated as a high frequency cut-off at a certain reduced frequency (b = 3/8) where those functions become significantly small (e.g., [6]). Accordingly, the background component in gust-front winds can also be expressed as:

D.K. Kwon, A. Kareem / Engineering Structures 48 (2013) 635–644

BG-F ¼ 1 

1 ½1 þ ðb  1220=HÞ2 1=3

ð14Þ

To investigate characteristics of the closed-form RMS response of fluctuating gust-front winds, three RMS responses, i.e., RMS response in the cases of boundary layer wind (B-L), approximated closed-form (TIFRF) and numerical analysis (TDFRF) are compared in terms of the following test conditions: B = D = 40 m, H = 200 m; natural frequency of a building n1 = 0.2 Hz (T1 = 5 s); building density qB = 180 kg/m3; q = 1.25 kg/m3; f1 = 0.01; V3-s = 40 m/s; Vmax = 56.68 m/s and zmax = 60.35 m; Exposure C in ASCE 7; Cx = Cz = 6; td = 10 and 200 s (td/T1 = 2 and 40, respectively). It can be observed that the closed-form solution cannot capture a transient response for very short pulse (Fig. 2a), while it becomes close to the numerical solution for a longer pulse duration (Fig. 2b).

639

Since the first modal frequency (x1) and its mode shape (/1) are given, a stiffness matrix (k) can be derived in terms of the orthogonality relationship of a mode shape, i.e., k/1 ¼ x21 m/1 . Note that alternatively, it is possible to invoke a modal analysis approach to obtain modal mass and stiffness for the first mode only. Similarly, a modal damping matrix for the first mode (C1) can be obtained a well-known relationship, C1 = 2f1x1M1, where M1 = a modal matrix for the first mode. For a multivariate simulation of wind fields, spectral representation method [26,27] is utilized with Davenport wind spectrum and coherence function [6] to generate wind fields for 50 nodal points in each simulation. A number of simulations are performed to properly represent the wind field and corresponding wind effects on the building. Probabilistic characteristics of the response are estimated through the statistical analysis of the simulated results.

4.2. Time domain analysis for nonstationary RMS response The gust-front factor framework including the generalized format in this study has been formulated in the frequency domain based on random vibration based analysis. In this section, a simulation-based approach in the time domain is introduced to assess the accuracy of the frequency domain approach, e.g., nonstationary fluctuating response, mean peak factor and extremes. Note that building and wind properties used here are those used in the previous section. 4.2.1. Structural modeling and wind simulation A lumped mass model with one degree of freedom at each level, i.e., a translational motion only, is utilized for structural modeling. In addition, the interstory height (dz) is assumed to be 4 m, which results in 50 nodal points along the height, and the mass matrix (m) is formulated by a lumped mass at each floor (=qB  B  D  dz).

Fig. 2. RMS displacements at building top: (a) td/T1 = 1 (zoomed plot); (b) td/T1 = 40.

4.2.2. RMS responses Fig. 3 shows a comparison of RMS displacement [rx(t)] (Fig. 3a) and velocity (rx_ ðtÞ) (Fig. 3b) response for a long pulse duration (td = 200 s) at the building top based on the analysis in the frequency and time domains, where the time domain results are obtained from 100,000 simulations. It is observed that there is a reasonably good match between the two approaches with small deviation in the displacement over a short interval. In a short pulse duration such as 5 s, it is very difficult to generate a reasonable wind field because it may not guarantee the accuracy of the match between the target and simulated wind spectra. In this short pulse case, the structural response may be considered similar to the one due to an impact load where free vibration after an impact may be dominant. In addition, it may be reasonable to assume that fluctuating wind effect in such a short time may be very small and negligible compared to the overall structural re-

Fig. 3. Comparison of RMS responses at building top by the time and frequency domain analyses in td = 200 s: (a) RMS displacement; (b) RMS velocity.

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sponses. Note that the time function used in the formulation is a half sine (Eq. (1)), thus it is expected that the freepvibration has a ffiffiffi form of a sinusoidal function with RMS equal to 1= 2. Accordingly, a time domain analysis is made without considering the fluctuating wind component. Fig. 4 shows a comparison of the RMS displacement and velocity response in time and frequency domains, which show good agreements. This confirms not only that the assumptions in the response analysis of a short pulse case was reasonable but also validates the accuracy of the frequency domain approach in comparison with the time domain.

4.2.3. Evaluation of the expected maximum of the nonstationary fluctuating response: td = 200 s One of the merits of the time domain analysis is that extreme distribution and consequently the expected maximum (mean extreme), which are of interest for wind design, can be directly obtained from a set of simulated data. Fig. 5 depicts cumulative density function (CDF) of the extreme using 100,000 simulated displacement peaks. Please note that total number of points shown are reduced in the plot for clear representation. It is observed that the lognormal and three-parameter generalized extreme value distributions (GEV) are the closest fits to the data-driven extreme value distribution. Note that the modified two-parameter Gumbel distribution as a function of the square of basic random variable (e.g., x2) provides a much better fit than conventional two-parameter Gumbel form, which agrees with the observation by Michaelov et al. [22,28]. The expected maximum is calculated as 0.0688 (m) in the time domain analysis. As alluded earlier, the expected maximum in the gust-front factor framework has been estimated using the multiplication of the mean peak factor (gG-F) and the maximum of time-dependent

Fig. 5. Comparison of extreme value distributions (GEV = three-parameter generalized extreme value distribution).

RMS response (max[rx(t)]) (e.g., Eq. (9b)) for simplicity, which is rewritten in the following:

max½~xG-F ðz; tÞ  g G-F  max½rx ðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:5772 g G-F ¼ 2 ln½2v 0 ðtÞT eq  þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ð15Þ

2 ln½2v 0 ðtÞT eq 

where Teq = an equivalent time; v0(t) = time-dependent zero upcrossing rate; qxx_ ðtÞ ¼ time-dependent correlation coefficient [22]:

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1  q2xx_ ðtÞ

rx_ ðtÞ  rx ðtÞ 2p r2xx_ ðtÞ qxx_ ðtÞ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r2x ðtÞr2x_ ðtÞ

v 0 ðtÞ ¼

ð16Þ

Michaelov et al. [22] showed that qxx_ ðtÞ approached zero within a short time, thus it can be approximated as qxx_ ðtÞ  0 and rx_ ðtÞ=½2prx ðtÞ  n1 yielding to m0(t)  n1. However, these assumptions are only valid when the nonstationary random excitation is white noise. Fig. 6 shows time-dependent zero upcrossing rate [v0(t)] for the example used in this section when td = 200 s and it is obvious that not only there exists a large variation in the range between 0 to td, but also the upcrossing rate does not converge to n1 (=0.2 Hz). For completeness, formulations of the time-dependent RMS responses [rx_ ðtÞ, rx(t) and rxx_ ðtÞ] needed to obtain timedependent correlation coefficient [qxx_ ðtÞ] for the evolutionary processes are given in Appendix A.Thus, a case study is performed to estimate the expected maximum with a set of different formulations and these values are compared with the baseline estimate, i.e., the value obtained in the time domain analysis: Case 1: Eq.

Fig. 4. Comparison of RMS responses at building top by the time and frequency domain analyses in td = 5 s: (a) RMS displacement; (b) RMS velocity.

Fig. 6. Time-dependent correlation coefficient (td = 200 s).

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D.K. Kwon, A. Kareem / Engineering Structures 48 (2013) 635–644 Table 2 Comparison of the expected maximum and sub-factor I3 for various cases.

The expected maximum (m) I3

Time domain

Case 1

Case 2

Case 3

Case 4

0.0688 0.8020

0.1004 (45.9%) 0.9012 (12.4%)

0.0901 (31.0%) 0.8687 (8.3%)

0.0907 (31.8%) 0.8708 (8.6%)

0.0814 (18.3%) 0.8414 (4.9%)

Note: the value in the parenthesis indicates a difference in percentage compared with the value in the time domain.

(15) with m0(t)  n1; Case 2: Eq. (15) with m0(t)  m0(td/2) where td/2 is the time when the time function (Eq. (1)) attains the maximum value; Case 3: gG-Freq with m0(t)  n1 where req = an equivalent RMS response [22]; Case 4: gG-Freq with m0(t)  m0(td/2). It is observed that overall, the expected maximum estimates in the frequency domain for the four cases are rather conservative as compared to the estimate in the time domain (Table 2). However, results suggest that such an implied conservatism in the estimated expected maxima using the frequency domain does not influence the overall picture, i.e., (generalized) gust-front factor. This is explained by the fact that as given in Table 2, the value of the subfactor I3 shows lesser variation from case to case. Thus, the conservatism in the estimated expected maxima is diffused by the lesser sensitivity of I3, which is proportional to (generalized) gust-front factor (Eq. (9)), to cases examined. Thus, the original format, Case 1, is utilized throughout this paper, however, for future refinements, more parametric studies will be required to arrive at a better representation. 4.3. Web-enabled portal based computation of gust-front factor In view of the complexity of the problem associated with nonstationary processes, a web-based portal to assess the design loads in gust-front winds in an e-design format (e.g., [29]) has been developed to facilitate and promote the usage of the gust-front factor in conjunction with ASCE 7 [1,2]. This would also permit onthe-fly evaluation of a number of loading cases which otherwise would require extensive calculation. It offers a fundamental framework that can be used at this stage and continually improved as additional information becomes available. This web-based on-line gust-front factor framework has a userfriendly interface and is available at http://gff.ce.nd.edu. Along with two profile criteria (Criterion 1 and 2), this portal offers a user-defined gust-front wind profile, i.e., Vmax and zmax based on Vicroy model (Eq. (2)). Thus, the user is not limited to utilizing the two profile criteria defined in this study, but also any arbitrary Vmax and zmax inputs for the on-line determination of gust-front wind load effects on buildings, which can be chosen without any restriction based by the ASCE 7 format. Once the input is completed, gust-front factor (GG-F), its sub-factors (I1 to I4) in Eq. (9a), and gust loading factor (GGLF) are then displayed in the results interface. Accordingly, one can also utilize this portal as a tool to evaluate generalized gust-front factor (GG,G-F) owing to the relationship between in GG,G-F and GG-F as shown in Eq. (9), i.e., I2 and (I3  GGLF), where gust loading factor (GGLF) can be obtained from a code and standard of interest, and consequently the design loads (ESWL) in Eq. (10) can be easily estimated.

C in ASCE 7 and terrain category II in Eurocode); other conditions are the same with the ones used in the previous section. 5.1. Asce 7 In this example, the example building is used to evaluate the GG,G-F, its associated ESWL (FG,G-F), ASCE 7-based ESWL under boundary layer winds (FASCE) and GG-F-based ESWL (FG-F) for the validation of GG,G-F approach. CD,G-F is assumed to be the same as conventional CD value, 1.3, and Criterion 2 at exposure C is used (Table 1). This results in GG,G-F = 1.77 (td/T1 = 2), 1.55 (td/T1 = 40) and GGLF = 1.82, which represent dynamic effects by gust-front winds to be of relatively lower significance than in conventional boundary layer winds. Nonetheless, one should not overlook possible load enhancement due to transient aerodynamics (CD,G-F), which may result in enhancing local pressures or overall force coefficient in the neighborhood of 5–20% based on preliminary observations thus far in comparison to conventional CD (e.g., [31,32]). In addition, the kinematic effects due to the wind profile [VG-F(z)] result in locally enhanced loads around zmax, therefore, this underscores the role of enhancement in the kinematic effects to the overall design loads (Fig. 7). For reference, the base shear and bending moments obtained from the ESWL based on FASCE

5. Examples of generalized gust-front factor In this study, two examples with codes and standards such as ASCE 7 [9] and Eurocode [30] are used to illustrate the GG,G-F, and associated ESWL (FG,G-F). Common parameters of the example building are: storm-translation speed (Vs-m) = 12 m/s; CD,G-F is assumed to be the same as conventional CD values, 1.3 (ASCE 7) or 1.5 (Eurocode) by assuming the transient aerodynamic effects, I4 in GG-F, is unity; Criterion 2 for open terrain condition, i.e., exposure

Fig. 7. Equivalent static wind loads (ESWL) by ASCE 7 (FASCE), gust-front factor approach (FG-F) and generalized gust-front factor approach (FG,G-F); (a) td = 10 s; (b) td = 200 s.

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D.K. Kwon, A. Kareem / Engineering Structures 48 (2013) 635–644

Table 3 Comparison of base shear and bending moment using ESWLs (FASCE and FG,G-F). td

Base forces

B-L (by FASCE)

G-F (by FG,G-F)

4

4

G-F/B-L (ratio)

10 s (td/T1 = 2)

Shear (kN) Bending moment (kN m)

1.51  10 1.56  106

2.52  10 2.37  106

1.67 1.52

200 s (td/T1 = 40)

Shear (kN) Bending moment (kN m)

1.51  104 1.56  106

2.22  104 2.08  106

1.47 1.34

and FG,G-F are compared in Table 3. It is noted that base forces in gust-front winds depending on pulse durations are 30–70% higher than those in boundary layer winds, which represent a significant increase due to gust-front effect in this example. This enhancement effect may be significantly influenced by the choice of gust-front velocity profile used, especially when more critical criterion such as Criterion 1 or AAFB microburst is used instead. Finally, it is observed that both ESWL by the gust-front factor (GG-F) and the generalized gust-front factor (GG,G-F) show precisely the same results which validates the accuracy of GG,G-F approach presented in this study. 5.2. Eurocode To demonstrate the application of the generalized gust-front factor approach to international codes and standards other than ASCE 7, Eurocode [30] is utilized in this example. Note that Eurocode utilizes 10-min mean wind speed for the basic wind speed (V10-min) thus this study utilizes an averaging time relationship suggested in the ISO standard [33]. The conversion relationship between V3-s and V10-min is V3-s = 1.46 V10-min, which results in V10-min = 27.4 m/s when V3-s = 40 m/s. In this application, one first needs to determine Vmax and zmax in the Vicroy model (Eq. (2)). This can be established following a modeling procedure described in Kwon and Kareem [1,2] in view of Eurocode: for brevity, Vmax and zmax given in Table 1 are utilized here. Once the determination of Vmax and zmax is made, next procedure is basically the same with the ASCE 7 case. For example, one needs to obtain gust loading factor (GGLF) using the Eurocode, which leads to the generalized gust-front factor (GG,G-F) (Eq. (9b)). The preceding procedure can be conveniently carried out without familiarity with random vibration based procedure presented here. Rather, as described in the earlier section, reliance can be made on the web-enabled portal for such computations with simple inputs. Finally, the design loads (ESWL) associated with GG,G-F in gust-front winds are estimated using Eq. (10). Fig. 8 shows the Eurocode-based ESWL under boundary layer winds (FEuro) and GG,G-F-based ESWL (FG,G-F) where FEuro is calculated based on the Eurocode. CD,G-F is assumed to be the same as conventional CD value, 1.5 in Eurocode. It is observed that overall trends are similar to the case of ASCE 7, i.e., dynamic effects caused by gust-front winds are of relatively less predominant than those by conventional boundary layer winds (GG,G-F = 1.76 in td/T1 = 2, 1.64 in td/T1 = 40 and GGLF = 2.034 in Eurocode) and the kinematic effects due to the wind profile [VG-F(z)] result in locally enhanced loads around zmax. 6. Discussion on gust-front loading effects on rigid structures Most codes and standards provide design wind loads for boundary layer winds based on gust loading factor approach for both flexible and rigid structures. In general, such a structure type, flexible or rigid, is generally classified based on the natural frequency of a structure. For example, a structure whose natural frequency is greater than or equal to 1 Hz are classified as a rigid structure. Unlike flexible structures where resonant component is pivotal, rigid

Fig. 8. Equivalent static wind loads (ESWL) by Eurocode (FEuro) and generalized gust-front factor approach (FG,G-F); (a) td = 10 s; (b) td = 200 s.

structures are mostly governed by static/quasi-static components such as the mean and the background. For example, ASCE 7 [9] defines gust loading factor for rigid structures as:

GGLF;rigid ¼ 0:925ð1 þ 1:7g Q Iz Q Þ

ð17Þ

where gQ = background peak factor (3.4 in ASCE 7); Iz = turbulence intensity; Q = background response factor. This equation is essentially the same as GGLF on flexible structures if the resonant response factor (R) is included. However, as alluded to in an earlier section, calculation of separate background and resonant components by the nonstationary fluctuating winds in gust-fronts may not always be possible unless long duration pulse exists (e.g., Eq. (11)). To investigate nonstationary fluctuating wind effects on rigid structures, Eq. (11) is utilized with the following example parameters for comparison purpose: B = D = 20 m; H = 40 m; td = 200 s; n1 = 2.0 Hz; f1 = 0.01 and 0.05, respectively; other parameters are the same with the previous example. Table 4 shows RMS background and resonant response components in boundary layer and gust-front winds for two damping ratios (f1). It is noted that the RMS response is very small for rigid structure because the dynamic effects are less dominant. Results in Table 4 suggest: (a) as expected, background response in both boundary layer and gustfront winds are insensitive to damping ratio, while the resonant response is reduced when the damping ratio increases; (b) back-

D.K. Kwon, A. Kareem / Engineering Structures 48 (2013) 635–644 Table 4 Comparison of background and resonant RMS responses on a rigid structure. Back. B-L (m) f1 = 0.01 f1 = 0.05

4

3.53  10 3.53  104

Res. B-L (m) 4

1.08  10 0.48  104

Back. G-F (m) 4

10.43  10 10.43  104

7. Concluding remarks

Res. G-F (m) 4

0.45  10 0.20  104

ground responses in gust-front winds are higher than those in boundary layer winds. This implies that wind profile effects (mean load effects) are more dominant in gust-front winds than in boundary layer winds. Consequently, the factor I3 (e.g., Eq. (9)) can be potentially larger than unity in the case of rigid structures; c) background responses are much larger than the resonant ones. It is because a wind spectrum decays fast with frequency after its peak and the spectral value in the natural frequency range of rigid structures is usually much smaller than that of flexible structures. This tendency is especially more dominant in gust-front winds, thus the effect of resonant response may be negligible without loss of accuracy in RMS response estimates of a rigid structure. Accordingly, for the evaluation of fluctuating wind effects on rigid structures where only background response component (B) is usually considered in boundary layer winds (e.g., Eq. (17)), it may be reasonable to use nonstationary RMS response without separately identifying the background response component in gust-front winds. An issue in the determination of the peak factor for rigid structures in gust-front winds still remains further examination. A conventional peak factor (gB-L) in Eq. (18) in boundary layer winds [6] is a function of zero upcrossing rate (m0) and observation time (T), e.g., T = 600 s (10-min) or 3600 s (1-h), where zero upcrossing rate m0  n1 for flexible structures when resonant response is dominant.

g B-L ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0:5772 2 lnð2v 0 TÞ þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 lnð2v 0 TÞ

643

ð18Þ

Most codes and standards in boundary layer winds distinguish the peak factor from the background (gQ) and resonant peak (gR) factors: gR is defined as in Eq. (18), while gQ is defined as a constant. For example, ASCE 7 utilizes gQ = 3.4, which originated from the extensive studies reported in Solari [7] and Solari and Kareem [34]. Please note that constant (gQ) is based on the assumption of an absolutely rigid structure (i.e., no resonance), thus Solari and Kareem [34] also suggested using a conservative value of 4.1 as the assumption may be practically unrealistic. While, the gust-front factor framework utilizes a mean peak factor in Eq. (15) [2,22] for the overall (background and resonant) RMS response, which is a similar form of a conventional peak factor (Eq. (18)). Even though the mean peak factor adopts an equivalent time (Teq), which is calculated based on the nonstationary RMS response [22] and generally much smaller than typical observation time used in boundary layer winds such as 10 min/1-h, the mean peak factor may lead to a conservative value in rigid structures if m0(t)  n1 is simply assumed, especially where n1 has a high natural frequency in the case of rigid structures. Thus, a plausible solution may be to directly calculate m0(t) and then apply to Eq. (15). For reference, some international codes and standards such as Architectural Institute of Japan [35], Eurocode [30] and National building code of Canada [36] etc. still utilize the format of Eq. (18) for rigid structures in boundary layer winds using a modified m0 instead. m0 is defined as n1C where C refers to a factor which is a function of the background and resonant responses [37]. m0(t) can be estimated from the RMS response (Eq. (16)) or alternatively, an approach used in boundary layer winds, i.e., based on wind velocity spectrum. Since background peak factor in gust-front winds is still unknown due to lack of information regarding the characteristics of gust-front winds, for practical purpose, it is tentatively suggested that the upper bound of the mean peak factor on rigid structures be limited to the background peak factor in boundary layer winds.

This paper introduces a generalized gust-front factor framework that captures dynamic features inherent in wind load effects in gust-fronts originating from a downburst/thunderstorm. This is akin to the gust loading factor format used in codes and standards world-wide for the treatment of conventional boundary layer winds. In the examples presented here, it is observed that as a result of gust-front winds in both ASCE 7 and Eurocode standards, a higher local ESWL distribution exists, despite the fact that the dynamic effects (GG,G-F) for the example buildings were lower than those in conventional boundary layer winds (GGLF). This underscores the role of enhancement in the kinematic effects introduced through the wind profile [VG-F(z)] in gust-fronts, which results in locally enhanced loads around zmax, to the overall design load. Moreover, one should not overlook the possible load enhancement due to transient aerodynamics (CD,G-F), which may result in enhancing local pressures or overall force coefficient in the neighborhood of 520% based on preliminary observations thus far in comparison to conventional CD. Additional research in the measurement of drag force in suddenly applied gust-fronts may further reinforce their initial finding. It is anticipated that the generalized gust-front factor presented here would experience further refinements over time similar to the many subsequent modifications/enhancements in conventional gust loading factor presented by Davenport [6], including its application to rigid structures even though the current format may still be acceptable for the case. For immediate design application, a computational framework of the gust-front factor is available in a user-friendly web-based portal (http://gff.ce.nd.edu). This promises to offer application of gust-front factor to international codes and standards and the flexibility of examining several loading configurations on-the-fly without actually becoming involved in detailed intensive computations for the evaluation of the generalized gust-front factor in terms of Eq. (9). This feature promises to make this as a valuable design tool that offers attractive features regardless of the user’s background in the various underlying computational aspects. It is worth noting that this framework relies on a series of simplifications and assumptions not only in an attempt to realistically capture the characteristics of gust-front winds but also to enable a closed-form solution, e.g., half-sine pulse wave, Vicroy Model, quasi-steady and strip hypothesis of the loading used in conventional gust loading/effect factor approach. These assumptions and hypothesis may not be fully representative of the actual complex gust-front winds with rapidly evolving wind field and attendant pressures/loads may differ spatially and temporally, e.g., effects of aerodynamics due to the opposing velocity gradients [31]. Experimental verification will be necessary in the future to either corroborate initial assumptions and validate the current framework or introduce modifications as deemed necessary. The web-enabled portal introduced here can be utilized for convenient modeling and as a tool for analysis/design. A preliminary framework has been introduced that is immediately applicable, but it also accommodates continual improvements as additional information becomes available. In particular, advances in understanding of gust fronts and the rational characterization of the flow field and its interaction with structures is necessary.

Acknowledgements The authors are grateful for the financial support provided in part by the NSF Grant CMMI 03-24331 and a collaborative research project between the NatHaz Modeling Laboratory and the Global

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Center of Excellence (GCOE) at Tokyo Polytechnic University funded by MEXT, Japan. Appendix A. Time-dependent RMS response A zero-mean modulated random excitation F(t) may be described by a multiplication of an amplitude modulating function A(t) and a stationary Gaussian process W(t):

FðtÞ ¼ AðtÞWðtÞ

ðA:1Þ

Then, a real-valued evolutionary process of a response x(t) can be expressed in the general form of a Fourier-Stieltjes integral [24,38,39]:

Z

1

Ax ðt; xÞeixt dZðxÞ Z t hðt  sÞAðsÞeixðtsÞ ds Ax ðt; xÞ ¼

xðtÞ ¼

1

ðA:2Þ

0

EfjdZðxÞj2 g ¼ Sww ðxÞdx where h(t) = unit impulse response; Sww(x) = power spectral den_ sity of W(t). Similarly, an evolutionary process of a response xðtÞ can be obtained from the differentiation of Eq. (A.2) as follows: Z 1 Z 1 _ xðtÞ ¼ ½A_ x ðt; xÞeixt þ ðixÞAx ðt; xÞeixt dZðxÞ ¼ A1 ðt; xÞdZðxÞ 1

A1 ðt; xÞ ¼

1

Z

t

_  sÞAðsÞeixðtsÞ ds hðt

0

ðA:3Þ _ Then, the evolutionary power spectral density (EPSD) of x(t) and xðtÞ can be expressed as:

Gxx ðxÞ ¼ jAx ðt; xÞj2 Sww ðxÞ Gx_ x_ ðxÞ ¼ jA1 ðt; xÞj2 Sww ðxÞ

ðA:4Þ

_ The evolutionary cross power spectral density of x(t) and xðtÞ, Gxx_ ðxÞ, is described as:

Gxx_ ðxÞ ¼ Ax ðt; xÞA1 ðt; xÞSww ðxÞ

ðA:5Þ

_ Accordingly, RMS responses of x(t) and xðtÞ can be obtained from the following relationship:

r2x ðtÞ ¼ r2x_ ðtÞ ¼

Z

1

Gxx ðt; xÞdx

Z1 1

Gx_ x_ ðt; xÞdx  Z 1  r2xx_ ðtÞ ¼ Im i Gx_ x_ ðt; xÞdx

ðA:6Þ

1

1

where Im = imaginary part of a complex. In this manner, timedependent correlation coefficient qxx_ ðtÞ in Eq. (16) can be estimated. References [1] Kwon DK, Kareem A. Gust-front factor: a new framework for the analysis of wind load effects in gust-fronts. In: 12th International conference on wind engineering (ICWE), Cairns, Australia; 2007. p. 767–74. [2] Kwon DK, Kareem A. Gust-front factor: new framework for wind load effects on structures. J Struct Eng, ASCE 2009;135(6):717–32. [3] Kwon DK, Kareem A. A framework for generalized gust-front factor. In: 5th European African conference on wind engineering (EACWE5), Florence, Italy; 2009.

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