Estimation of multiple limit state responses with various mean recurrence intervals considering directionality effects

Journal of Wind Engineering & Industrial Aerodynamics 193 (2019) 103959

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Estimation of multiple limit state responses with various mean recurrence intervals considering directionality effects Min Liu a, *, Xinzhong Chen b, a, Qingshan Yang a, c a b c

School of Civil Engineering, Chongqing University, Chongqing 400044, China National Wind Institute, Department of Civil, Environmental and Construction Engineering, Texas Tech University, Lubbock, TX, 79409, USA Beijing’s Key Laboratory of Structural Wind Engineering and Urban Wind Environment, Beijing, 100044, China

A R T I C L E I N F O

A B S T R A C T

Keywords: Extreme value distribution Directionality effect System reliability Extreme wind load effects Cladding on saddle type roof

Assessment of system reliability of structures against wind loading requires evaluation of the joint probability distribution of multiple limit state responses with consideration of wind and aerodynamic directionality effects. While the directionality effect has been extensively addressed in literature on the estimation of probabilistic single limit state response, it is not the case for multiple limit state responses. This study revisits estimation of extreme responses with various mean recurrence intervals, with a focus on adequate definition of directionality factor. A new framework is then presented for estimation of joint probability distribution of multiple extreme responses with consideration of directionality effect. This new framework can be considered as an extension of existing efforts on directionality effect for single limit state response to multiple ones, thus help in better assessing system reliability of structures against wind loading. The proposed framework is applied to responses of claddings on a large-span saddle type roof.

1. Introduction Performance-based design of structures against wind loading requires estimation of wind load effects (responses) with various mean recurrence intervals (MRIs) by taking into account the probability distribution and directionality of wind speed and wind induced response. Cook and Mayne (1979, 1980) proposed a probabilistic method for calculating the distribution of maximum response of rigid structures using the probability distributions of yearly maximum wind speed and extreme response coefficient conditional on wind speed. It is often referred to as the first-order method because it does not consider the probability of yearly extreme response produced by the second and higher-order largest wind speeds in a year. This probability of occurrence can be further considered in so-called full-order method (Harris, 1982, 2005; Chen and Huang, 2010). Chen (2015) presented a closed-form formulation for estimating responses with various MRIs which sheds more physical meaning and is easy for practical application. Other probabilistic wind load effect models have also been used in literature which often involve Monte Carlo simulations (MCS) for the quantifications (e.g., Kareem, 1987; Diniz and Simiu, 2005). On the other hand, a number of approaches have been developed to address wind directionality effect. The random process crossing rate

approach using the parent distribution of directional mean wind speed has been widely used (Davenport, 1977; Wen, 1983, 1984; Simiu and Scanlan, 1996). Improvements of this approach have been made by using bi-modal Weibull distribution of wind speed (Isyumov et al., 2014), and a mixed distribution with more accurate modeling of distribution tail (Zhang and Chen, 2016a). Another method is directly based on the historical directional yearly maximum wind speed data (Simiu and Filliben, 1981; Simiu and Heckert, 1998). The storm passage method was introduced in Isyumov et al. (2003) which directly used time history of directional mean wind speed during a storm such as Hurricanes. Method of using directional extreme winds in mixed wind climates was developed in Matsui and Tamura (2004). The simple sector-by-sector approach has two variants where the directional wind speeds are approximated as independent or fully correlated. The aforementioned approaches for directionality effect often treat the extreme response coefficient as a deterministic quantity. The influence of its uncertainty on directionality effect was discussed in Laboy-Rodríguez et al. (2014) based on time histories of directional wind speed. Zhang and Chen (2015) developed an approach for estimating probabilistic response with consideration of both directionality and uncertainty, where the joint probability distribution of directional extreme wind speeds was modeled by multivariate extreme value theory with Gaussian copula. The influences of directional

* Corresponding author. E-mail address: [email protected] (M. Liu). https://doi.org/10.1016/j.jweia.2019.103959 Received 24 January 2019; Received in revised form 5 July 2019; Accepted 19 July 2019 Available online 31 July 2019 0167-6105/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Probability distribution of yearly maximum wind speed in each direction.

Fig. 2. Directional extreme wind speeds with different MRIs and the COV in each direction.

dependence, model difference and wind speed masking were further investigated in Zhang and Chen (2016b). The correlation between directional wind load effects was also considered by Warsido and

Bitsuamlak (2015). In structural design, the directionality effect is often accounted by a directionality factor, which permits a reduction of extreme response with 2

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probability distribution of multiple extreme responses with consideration of directionality effects of wind speed and aerodynamics. The present study can be considered as an extension of existing efforts on directionality effect for single limit state response to multiple ones, thus helps in better assessing system reliability of structures against wind loading. The proposed framework is applied to extreme responses of claddings on a large-span saddle type roof.

Table 1 Correlation coefficient matrix of directional wind speeds. Dir.

N

NE

E

SE

S

SW

N NE E SE S SW W NW

1 0.50 0.09 0.52 0.63 0.57 0.13 0.78

1 0.13 0.37 0.25 0.27 0.15 0.28

1 0.38 0.01 0.08 0.19 0.09

1 0.23 0.41 0.18 0.40

1 0.52 0.02 0.64

W

NW

1 0.05

1

Sym. 1 0.18 0.34

2. Methodology 2.1. Probability distribution of a yearly maximum response The extreme response of interest under the mean wind speed at i th direction, determined from wind tunnel test or structural analysis, can be expressed as wi ¼

1 2 ρv ci ðvi Þ 2 i

(1)

where ρ is the air density which can be regarded as a constant under strong wind speed; vi is the mean wind speed at i th direction; ci ðvi Þ is the extreme response coefficient, which can be independent of wind speed or a function of wind speed. When extreme response coefficients are considered as deterministic values, the cumulative distribution function (CDF) of the yearly maximum response considering all directions can be readily determined as Fig. 3. Dependence structure pattern of yearly maximum wind speeds in N and NW directions.

Ψ W ðwÞ ¼ FV ðv1w ; v2w ; ⋯vnw Þ

(2)

where viw ði ¼ 1; 2; …; nÞ is the mean wind speed at i th direction causing pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi response level w, which is calculated by Eq. (1) as viw ¼ 2w=ρci ðviw Þ; and FV ðv1w ; v2w ; ⋯vnw Þ is the joint CDF (JCDF) of yearly maximum directional wind speeds. The MRI of response level w is R ¼ 1=ð1  Ψ w ðwÞÞ. When the extreme response coefficients are considered as random variables, the CDF of yearly maximum response can be calculated as (Zhang and Chen, 2015) Z

Z

Ψ W ðwÞ ¼

Fig. 4. Distribution wind direction.

of

yearly

maximum

wind

speed

regardless

⋯

FC ðc1 jv1 ; c2 jv2 ; …; cn jvn ÞfV ðv1 ; v2 ; ⋯vn Þdv1 dv2 ⋯dvn

(3)

where fV ðv1 ; v2 ; ⋯vn Þ is the joint probability density function (JPDF) of directional yearly maximum wind speed; FC ðc1 jv1 ; c2 jv2 ; …; cn jvn Þ is  JCDF of response coefficients and ci vi ¼ 2w=ρv2i ði ¼ 1; 2; …; nÞ: The above multiple integration can be estimated by MCS as follows with enhanced computational efficiency. The samples of vi ði ¼ 1; 2; …; nÞ are firstly generated based on their JPDF, the corresponding FC ðc1 jv1 ; c2 jv2 ; …; cn jvn Þ is then calculated. The ensemble average of FC ðc1 jv1 ; c2 jv2 ; …; cn jvn Þ gives Ψ W ðwÞ. It is generally assumed that ci jvi ði ¼ 1; 2; … ; nÞ are mutually independent (Zhang and Chen, 2015), thus FC ðc1 jv1 ; c2 jv2 ; …; cn jvn Þ ¼ Ψ C1 ðc1 jv1 ÞΨ C2 ðc2 jv2 Þ ⋯Ψ Cn ðcn jvn Þ, where Ψ Ci ðci jvi Þ is the CDF of ci ðvi Þ. When the directional wind speeds are mutually independent, Eqs. (2) and (3) become:

of

a given MRI from that of the worst case estimation (Simiu and Filliben, 1981; Habte et al., 2014). Such a directionality factor is sensitive to the directionality characteristics of wind speed and aerodynamics (Isyumov et al., 2014; Zhang and Chen, 2015 and 2016b). A directionality factor of 0.85 is introduced in ASCE 7-10. On the other hand, the directionality effect was introduced by using directional design wind speeds (Matsui and Tamura, 2004; Australian/New Zealand Standard AS/NZS 1170.2:2002). Evaluation of structural system performance against wind loads requires estimations of multiple probabilistic limit state responses and their correlations. The existing approaches estimate these multiple limit state responses individually without further investigation of their statistical dependence. These estimations do not provide sufficient information regarding the overall structural system reliability associated with multiple limit state responses (Allsop, 2011). In this study, after reviewing different approaches of treating directionality effect for single limit state response in design codes and standards, an alternative approach of defining directionality effect is proposed. Besides, a framework is introduced for estimating the joint

Ψ W ðwÞ ¼ Ψ V1 ðv1 ÞΨ V2 ðv2 Þ⋯Ψ Vn ðvn Þ

(4)

and Ψ W ðwÞ ¼ ΨW1 ðwÞΨW2 ðwÞ⋯ΨWn ðwÞ Z ΨWi ðwÞ ¼

0

∞

Ψ Ci ðci jvi ÞϕVi ðvi Þdvi

(5) (6)

where ϕVi ðvi Þ and Ψ Vi ðvi Þ are PDF and CDF of yearly maximum wind speed in i th direction; and ΨWi ðwÞ is CDF of yearly maximum response

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Fig. 5. Pressure measurements of a saddle roof model in wind tunnel.

wind speed in ith direction are fully correlated, the JCDF of multiple responses coefficients is

associated with wind speed in the i th direction, which can be calculated in closed-form formulation (Chen, 2015). In practice, the following simplified sectorial approach is often used for estimating R-year wind load effect. The R-year response at each wind direction is calculated, i.e., Ψ Wi ðwRi Þ ¼ 1  1=R, where wRi is R-year response in i th direction. Their maximum over all directions is approximately considered as the R-year wind load effect with consideration of directionality, i.e., wR ¼ maxðwR1 ; wR2 ; …; wRn Þ. As pointed out in Simiu and Filliben (2005), this sectorial approach can lead to an underestimation.

FCij jvi ðci1 jvi ; ci2 jvi ; ⋯; cim jvi Þ ¼ ΨCik jvi ðcik Þ

where Ψ Cik jvi ðcik Þ is CDF of dominant response coefficient cik jvi . For the second case when the multiple extreme responses under the same wind speed and direction are considered to be mutually independent, we have FCij jvi ðci1 jvi ; ci2 jvi ; ⋯; cim jvi Þ ¼

The multiple responses wðjÞ ðj ¼ 1; 2; …; mÞ can further be investigated, where m is the total number of wind induced responses that are considered for estimation of joint probability distribution. The multiple responses are functions of wind speed and direction expressed similar to Eq. (1). For example, the wind speed in ith direction which causes wðjÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi wj is denoted as viwj ¼ 2wj =ρcij ðvij Þ, where cij is response coefficient related to jth response in ith wind direction. When the response coefficients are considered as deterministic values, the JCDF of yearly maximum multiple responses is determined as

FW ðw1 ; w2 ; ⋯; wm Þ ¼

Z

FC1j jv1 ðc11 jv1 ; c12 jv1 ; ⋯; c1m jv1 Þ⋯

Yn i¼1

Z FWi ðw1 ; w2 ; ⋯; wm Þ ¼

  ΨCij jvi cij

(10)

FWi ðw1 ; w2 ; ⋯; wm Þ

(11)

0

∞

Ym j¼1

  ΨCij jvi cij ϕVi ðvi Þdvi

(12)

Once the CDF of single yearly maximum response can be calculated by section 2.1., the JCDF of multiple yearly maximum responses can also be calculated by separate CDFs of multiple responses under the assumptions of full correlation and independent. For full correlation, the JCDF is calculated by

(7)

  FW wð1Þ ; wð2Þ ; ⋯; wðmÞ ¼ Ψ wðjÞ ðwÞ

(13)

where Ψ wðjÞ ðwÞ is CDF of the dominant response among multiple responses. For independent case, the JCDF is calculated by

Z ⋯

j¼1

where

where vimin ¼ minðviw1 ; viw2 ; …; viwm Þ: When the uncertainties of response coefficients are further considered, and assuming that the response coefficients caused by wind speed at different directions are statistically independent, the JCDF of multiple yearly maximum responses is calculated as FW ðw1 ; w2 ; ⋯; wm Þ ¼

Ym

 where Ψ Cij jvi ðcij Þ is CDF of response coefficient cij vi . When the directional wind speeds are mutually independent, and the responses at same wind direction are also mutually independent, we have:

2.2. Joint probability distribution of multiple yearly maximum responses

FW ðw1 ; w2 ; ⋯; wm Þ ¼ FV ðv1min ; v2min ; …; vnmin Þ

(9)

(8)

FCnj jvn ðcn1 jvn ; cn2 jvn ; ⋯; cnm jvn ÞfV ðv1 ; …; vn Þdv1 ⋯dvn

  Ym FW wð1Þ ; wð2Þ ; ⋯; wðmÞ ¼ Ψ ðjÞ ðwÞ j¼1 w

where FCij jvi ðci1 jvi ; ci2 jvi ; ⋯; cim jvi Þ is the JCDF of multiple response co efficients cij vi ¼ 2wj =ρv2i ði ¼ 1; 2; …; n; j ¼ 1; 2; …; mÞ. The MCS is adopted for the calculation similar to the calculation of Eq. (3). There are two particular cases exist for the multiple response coefficients under a given wind direction including fully correlated and mutually independent. Thus there are two particular cases for the general solution of Eq. (8). For the first case, when the responses associated with

(14)

where Ψ wðjÞ ðwÞ is the CDF of single yearly maximum response.

2.3. Estimation of system reliability After the probability distributions of wind responses are derived, the 4

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Fig. 6. Contours of negative peak pressure coefficients.

2.4. Joint probability distribution of multiple extremes by Gaussian Copula

failure probability (i.e., exceedance probability) of system with multiple wind responses can be calculated through incorporating the probability distributions of resistance by Z Pf ¼ 1 

The Gaussian copula can be used to model the joint probability distribution of multiple extremes, which is applied to the directional yearly maximum wind speeds and multiple extreme response coefficients. The Gaussian copula is defined as

Z ⋯

FW ðr1 ; r2 ; ⋯; rm ÞfR ðr1 ; r2 ; ⋯; rm Þdr1 ⋯drm

(15)

where fR ðr1 ; r2 ; ⋯; rm Þ is the JPDF of resistance, which can be calculated by fR ðr1 ; r2 ; ⋯; rm Þ ¼ fR1 ðr1 Þ⋯fRm ðrm Þ when the correlation between the resistances is disregarded. The multiple integration in Eq. (15) can be estimated by MCS. The MRI is then estimated as 1 =Pf years. The reliability index β ¼ Φ1 ð1  Pf Þ, where Φ is the CDF of standard Gaussian distribution.

FX ðx1 ; x2 ; …; xn Þ ¼ GY ðy1 ; y2 ; …; yn Þ

(16)

where FX ðx1 ; x2 ; …; xn Þ is the JCDF of multiple extremes X1 ; X2 ; …; Xn ; GY ðy1 ; y2 ; …; yn Þ is the JCDF of Gaussian variables Y1 ; Y2 ; …; Yn with zero mean vector and covariance matrix Σ with Σ ii ¼ 1 and Σ ij ¼ Σ ji ¼ ρij , which is the correlation coefficient between Yi and Yj . The extreme variable Xi and standard Gaussian variable Yi are related by the mono-

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Fig. 9. Extreme value distribution of internal force coefficient of screw S2.

   xi  mi ΨXi ðxi Þ ¼ exp  exp  δi

(18)

where mi and δi are mode and dispersion coefficients. The mean and STD pffiffiffi are μXi ¼ mi þ γδi and σ Xi ¼ δi π = 6; where γ ¼ 0.5772 is the Euler constant. Weibull (Type III) extreme value distribution can also be used while it is less conservative as compared to Gumbel (Type I) distribution (e.g. Kasperski and Hoxey, 2008). The translation function is thus calculated as

Fig. 7. Layout and dimension of steel cladding (Unit: mm).

gi ðyi Þ ¼ 

pffiffiffi 6

π

ðInf  In½Φðyi Þg þ γÞ

(19)

It is noted that the translation function thus the relationship between the correlation coefficients of the Gumbel distribute variable and standard Gaussian variable is not affected by the distribution parameters of Gumbel distribution (Liu and Der Kiureghian, 1986; Zhang and Chen, 2015). The relationship can be approximated as following (Liu and Der Kiureghian, 1986)

ρij ¼ ζij 1:064  0:069ζij þ 0:005ζ2ij

Fig. 8. Continuous multiple span beam model of cladding (Unit: mm).

 (20)

3. Extreme response of claddings on a saddle type roof Table 2 Influence coefficients of internal forces for screws S1, S2, S3 and S4 under Taps 87, 86 and 85. Influence coefficients

Cs1

Cs2

Cs3

Cs4

Isi1 Isi2 Isi3

0.83 0.06 0.03

0.81 1.56 0.16

0.16 1.56 0.81

0.03 0.06 0.83

3.1. Directional extreme wind speed model The meteorological data in this study were obtained from one of the Beijing Meteorological Stations recorded from January 1st, 1951 to December 31st, 2007, which is governed by China Meteorological Administration. The data are open to public and can be downloaded from http://data.cma.cn/data/detail/dataCode/SURF_CLI_CHN_MUL_DAY _CES.html. The measurement height of the station is 10 m. The average time of mean wind speed is 10 min. The directional wind speeds are divided into 16 directional sectors. The daily maximum mean wind speed and the corresponding wind direction are provided in the data set. After post-processing, 8 sectors are kept representing the directions of N, NE, E, SE, S, SW, W and NW. Another reason for choice of 8 sectors is that only 8 angles of attack are carried out in wind tunnel tests of this study. When the wind load effects are sensitive to wind direction, a much finer resolution of directional sectors should be adopted. Since only daily maximum wind speed rather than 10-min mean wind speed is provided in the dataset, the “masking” problem exists. Two directional yearly maximum wind speed data were completely masked, which are approximately calculated from neighboring data (e.g., Payer and Küchenhoff, 2004; Zhang and Chen, 2016b). The masked yearly maximum wind speed in the direction γ 1 , i.e., vγ1 , is estimated as vγ1 ¼ vγ2 cosðγ 1  γ 2 Þ, where jγ 1  γ 2 j < π =2, and vγ2 is the recorded mean wind speed at the direction γ 2 .

tonic translation function ðxi  μXi Þ =σ Xi ¼ Ψ 1 Xi ðΦðyi ÞÞ ¼ gi ðyi Þ, where μXi and σ Xi are the mean and standard deviation (STD) of Xi ; Ψ Xi ðxi Þ and Φðyi Þ are CDFs of Xi and Yi , respectively. The correlation coefficient between Xi and Xj , i.e., ζij ¼ ½EðXi Xj Þ 

μXi μXj =σ Xi σ Xj is related to that of Gaussian variables Yi and Yj as (Grigoriu, 1995). Z ζ ij ¼

∞

∞

Z

∞

∞

    gi ðyi Þgj yj φ yi ; yj ; ρij dyi dyj

(17)

where φðyi ; yj ; ρij Þ is the JPDF of bivariate Gaussian distribution. Some properties of ζij have been proved and given in Liu and Der Kiureghian (1986). In this study, Gumbel (Type I) distribution is used for modeling the distribution of Xi , i.e.,

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Fig. 10. Mean and COV of internal force coefficients of screws S1, S2, S3 and S4.

W and SW directions are quite close and are higher than those in SE, E and NE directions. The COVs of directional yearly extreme wind speeds range from 20% to 36%. The largest COV happens in W direction. Table 1 gives the covariance matrix calculated from yearly maximum wind speed data. The covariance matrix of underlying Gaussian variables is very close to that of wind speed according to Eq. (20). The largest correlation coefficient is between wind speeds in N and NW, which is 0.78. Fig. 3 shows the contours of JPDFs of wind speeds in N and NW directions, and their underlying Gaussian variables. It is observed that elliptical shape can portray the dependence relationship between two underlying Gaussian variables thus the directional wind speeds. The CDF of directionless yearly maximum wind speed is determined using the multivariate extreme wind speed model according to Eq. (16), which is denoted by Ψ V ðvÞ in Fig. 4. It can also be determined from the non-directional maximum wind speed data directly, denoted as Ψ V;non in Fig. 4. Both are quite consistent at lower MRI while some differences are observed as higher MRIs, which are attributed to modeling uncertainty due to limited wind speed samples and inconsistency of asymptotic extreme value distribution (Zhang and Chen, 2015). The distribution is also determined by assuming directional wind speeds are mutual independent or fully correlated. At higher MRIs, the CDF of directionless yearly maximum wind speed is close to that of wind speed in N direction. The 50- and 500-year directionless wind speeds by multivariate model are 27.3 and 35.3 m/s, respectively. The 50- and 500-year wind speeds in N direction are 26.9 and 35.1 m/s, respectively. The correlation of directional wind speeds do not noticeably affect the distribution of wind speed regardless of wind direction because the wind speed in N direction is dominant.

Fig. 11. Contour of joint PDF of extreme internal force coefficients of screws S2 and S3 at angle of attack of 0 degree.

3.2. Directional extreme response coefficients The extreme wind effects of steel cladding on a saddle type roof are addressed in this study. The wind pressure coefficients data on the saddle type roof were collected from a wind tunnel test at Beijing’s Key Laboratory of Structural Wind Engineering and Urban Wind Environment in Beijing Jiaotong University in China (Liu et al., 2016). The saddle type roof model is featured with a square platform with a side length of 60 cm, two low points with a height of 10 cm, and two high points with a height of 24.1 cm as shown in Fig. 5(a). A total of 265 pressure taps are distributed on the roof. The length ratio of the model to the prototype is 1:100, and scaling ratio of wind speed is 1:2. Therefore, the time ratio is 1:50. The mean wind speed profile is approximated by a power law with an exponent of 0.10. The reference height is 21 cm. The longitudinal turbulence intensity and turbulence integral scale at the reference height are 9.28% and 35 cm, respectively. Three wind directions are tested as shown in Fig. 5(b). At each testing direction, there are 270 sets of wind pressure records with a sampling frequency of 312.5 Hz and a time

Fig. 12. Correlation coefficient of extreme internal force coefficients of screws S2 and S3 as function of angle of attack.

There are 57 yearly maximum wind speed data available in each sector, which are fitted into Gumbel distribution directly. The wind speed data are ranked in ascending order and the probability of non-exceedance of ith wind speed is then calculated by ði  0:44Þ=（N þ 0:12Þ, where N is the number of total yearly maximum wind speed samples (Holmes, 2015). Fig. 1(a)-(h) show the fitting of CDF of yearly maximum wind speed at 8 directions. Fig. 2 gives the 50- and 500-year wind speeds and the COV of yearly maximum wind speeds in these wind directions. It is observed that the wind speed in N direction is largest, and those in NW,

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Fig. 13. Directional internal forces of screw S2 for structural orientation angle of 0, 90, and 315 degrees.

Fig. 14. 50- and 500-year internal forces of screw S2 for different structural orientations.

height of 21 m; D ¼ 250mm is the screw spacing as shown in Fig. 7 (b). When calculating the force coefficient, the mean wind speed at 10 m height needs to be concerted to that at the reference height of 21 m using the power law profile of mean wind speed. The cladding affected by the pressure taps at different regions are investigated including at the corner (Taps 3, 21, 39), near the edge (Taps 87, 86, 85, and Taps 16, 34, 52), and in the middle of roof (Taps 253, 254). The internal force coefficient Csi is calculated as

duration of 12 s in model scale (10 min in prototype). The dynamic pressures are given as pressure coefficients normalized using the mean wind speed of 9.21 m/s at the reference height. The negative peak pressure coefficients contours for three testing directions are shown in Fig. 6. More details on the statistics of pressure coefficients can be found in Liu et al. (2016). The high-strength trapezoidal steel cladding is used as the roof sheathing in this study. The size of a single cladding is 750 mm  6000 mm with a thickness of 0.6 mm. The layout of the cladding on the roof is shown in Fig. 7(a). The dimension of cross section is shown in Fig. 7(b). The cladding is connected on the purlin by screws at alternate crests. The internal forces of screws can be calculated based the assumption of continuous multi-span beam with a span length of 2000 mm under uniform wind loads as shown in Fig. 8 (Henderson and Ginger, 2011). The internal force of ith screw is represented by the force coefficient Csi ¼ Fsi =ð0:5ρV 2 D), where V is the wind speed at reference

Csi ¼ q1 Isi1 þ q2 Isi2 þ q3 Isi3

(21)

where q1 , q2 and q3 are the uniform loading coefficients obtained from wind pressure measurements; Isi1 , Isi2 and Isi3 , are the influence coefficients, which are listed in Table 2 for screws under Taps 87, 86 and 85. The influence coefficients for the screws under other taps are not listed for brevity. In the following, unless specified, the screws are

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Fig. 15. Directionality factors of different internal forces as functions of structural orientations.

Table 3 Equivalent MRI, wind speed, and estimation deviations of screw internal forces. Wind directions

α ¼ 0 (Taps 87, 86, 85) α ¼ 90 (Taps 87, 86, 85) α ¼ 315 (Taps 87, 86, 85) α ¼ 135(Taps 3, 21, 39) V E (m/s)

TE VE TE VE TE VE TE VE

(years) (m/s) (years) (m/s) (years) (m/s) (years) (m/s)

N

NE

E

SE

S

SW

W

NW

Error

98

2.1e5

2.4e9

1.2e7

3.0e4

6.8e8

559

5.4%

50 26.9 136 30.5 1.5e5

7.9e3

7.2e6

3.1e6

1.4e7

3.4e7

9.8e4

89 24.6 1.3e8

9.8e5

7.2e7

3.3e4

9.7e6

185

247

149

14.4%

1.5e4

2.8e4

2.7e3

1.7e3

8.4e7

7.4e6

11.0%

27.6

18.6

14.7

52 16.1 17.0

19.2

18.6

18.9

24.0

4.8%

Fig. 12. Strong correlation is observed at angles of attack of 0, 45, 90 and 135 . The internal force coefficients are considered to be independent of wind speed.

referred to those under Taps 87, 86 and 85. The time history of internal force coefficient of each screw is calculated, from which 270 samples of extremes are derived and are fitted into a Gumbel distribution. For example, Fig. 9 shows the results for the screw S2 at angle of attack of 0, 45 and 90 . The results at angles of attack of 135, 180, 225, 270 and 315 are derived by using the symmetry of the testing model. Fig. 10 plots the mean and COV of extreme internal force coefficients of screws S1, S2, S3 and S4 at different angles of attack. It can be seen that S2 and S3 have larger mean extreme internal force coefficients. Among all angles of attack, the largest value happens under 270 . The COVs of S2 and S3 are smaller than those of S1 and S4. The largest COV of 29.2% is observed at angle of attack of 135 . The JPDF of the extreme internal force coefficients of two screws at each angle of attack is determined by Gaussian copula. Fig. 11 presents the contour of JPDF of extreme internal force coefficients of screws S2 and S3 under angle of attack of 0 whose correlation coefficient is 0.89. The correlation coefficients under different angles of attack are shown in

3.3. Estimation of screw internal forces with various MRIs 3.3.1. Influence of structural orientation The structural orientation is defined by an angle α, which is the angle between N wind direction and zero angle of attack, and increases with structural rotation clockwise. Accordingly, the structural orientation with an angle α indicates that N wind direction aligns with the angle of attack of 360  α degrees. The structural orientation varies from 0 to 360 in every 45 . For each structural orientation, the extreme value distributions of the internal forces of screws S1, S2, S3 and S4 are calculated with consideration of both directionality and uncertainty using Eq. (3). In this section only the results of S2 under Taps 87, 86 and 85 are presented for brevity, which has the largest internal force. Fig. 13 9

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Fig. 14 displays the estimated 50- and 500-year internal forces with consideration of directionality and uncertainty, as functions of structural orientation. The results at structural orientations of 0, 90 and 315 are also shown in Fig. 13. The 50- and 500-year responses have similar dependence on structural orientation. The normalized 500-year response shown here is in fact the directionality factor (Simiu and Scanlan, 1996). For the purpose of comparison, the estimations by assuming the directional winds are fully correlated and mutually independent are also presented. It can be seen that the results with consideration of correlation of directional wind speeds are between those of fully correlated and independent cases. When there is only one dominant wind direction, say, at structural orientation of 90 , the correlation of directional wind speed has no influence. When there are several dominant directions, say, at structural orientation of 0 and 315 , the correlations between dominant wind speeds show some influences. Due to the relatively low correlation between directional wind speeds as shown in Table 1, the results with consideration of correlation of directional wind speeds are almost the same with that of independent case. In another word, for the wind climate model addressed in this case study, the directional extreme wind speeds can be considered as independent.

Fig. 16. Contour of JCDFs of internal forces of screw S2 and S3 for α ¼ 0 (Taps 87, 86, 85).

3.3.2. An alternative approach of defining directionality effect The directionality factor (Kd ) of internal force of S2 is replotted as function of structural orientation in Fig. 15 (a). The directionality factor of 50- and 500-year responses are almost same, ranging from 0.43 to 1 for different structural orientations. The directionality factor averaged over all structural orientation is 0.65, which is considered as the directionality factor when building orientation is unknown and follows a uniform distribution over 0–360 . The directionality factors of S2 in other regions of taps are also plotted in Fig. 15. The averaged directionality factors over structural orientation are 0.65, 0.52, 0.60 and 0.76, respectively. The variation character of directionality factor depends on individual response. In Australian/New Zealand standard (AS/NZS 1170.2:2011), the sector-by-sector approach is adopted instead of using a single directionality factor. The wind speed in each direction for a given MRI of R years is defined. To estimate the R-year response, the response at each direction caused by R-year wind speed is determined. The maximum response over all directions is then taken as the R-year response. As pointed out in Simiu and Filliben (2005), the sector-by-sector approach can be unconservative. The accuracy of this approach is examined here as applied to previously discussed example, i.e., internal force of screw S2 under Taps 87, 86 and 85 at different structural orientations. As shown in Fig. 15(a), at structural orientation of 0, 90 and 315 , the 500-year responses are 0.73, 0.99 and 0.57, respectively. The sector-by-sector approach gives 0.66, 0.99 and 0.48, having underestimation deviations of 17.4%, 0% and 15.8%, respectively. For the 50-year responses, the underestimation deviations are 11.3%, 0% and 20.0%, respectively. For structural orientation of 90 , the sector-by-sector approach is accurate enough since there is only one dominant wind direction. The Architectural Institute of Japan (AIJ) recommendations (AIJ-RLB-2004) adopted a different approach to account for directionality effect. The R-year response is estimated as the maximum response

Fig. 17. Contour of JCDFs of internal forces of screw S2 and S3 for α ¼ 180 (Taps 253, 254).

shows the 50- and 500-year internal forces in each wind direction for the structural orientation angle of 0, 90, and 315 . The results are normalized by 500-year internal force without consideration of directionality effect, i.e., worst case estimation, which is calculated by combining the probability distribution of yearly maximum wind speed regardless of wind direction and the distribution of internal force efficient at the most sensitive direction (angle of attack of 270 for S2). The 50- and 500-year internal forces without the directionality effect are 739 N and 1236 N. It can be observed that there is only one dominant direction for structural orientation angle α ¼ 90 , i.e., N direction, while there are two and four dominant directions, for α ¼ 0 and 315 , i.e., N and NW directions, and N, NW, W and SW directions, respectively.

Table 4 Reliability index and MRI of single screw S2 for different structural orientations (deterministic response coefficients). Orientation α (degrees), β and MRI (years) Taps 87, 86, 85

α ¼0 α ¼ 90 α ¼ 315

Taps 3, 21, 39

α ¼ 135

β MRI β MRI β MRI β MRI

Multivariate model, Eq. (2)

AS/NZS 1170.2:2011

AIJ-RLB-2004

ASCE 7-10

Alternative method in section 3.3.2

2.08 53 1.73 24 2.73 316 2.37 113

2.28 88 1.80 28 2.93 592 2.39 120

2.17 67 1.70 22 2.98 686 2.22 76

1.90 35 1.90 35 1.90 35 0.87 5

2.00 44 1.64 20 2.71 294 2.37 113

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Table 5 Reliability index and MRI of single screw and system with multiple screws under Taps 87, 86 and 85 with different structural orientations (response coefficients treated as random variables). Orientation α (degrees), β and MRI (years)

Screw S2

Screw S3

System Real-Eq. (8)

Full correlation, directional-Eq. (9)

Independent, directional-Eq. (10)

Full correlation-Eq. (13)

Independent-Eq. (14)

α ¼ 0

2.04 48 1.69 22 2.42 130

2.08 53 1.79 27 2.49 156

1.99 43 1.65 20 2.38 116

1.99 43 1.65 20 2.40 121

1.91 36 1.65 20 2.34 105

2.04 48 1.69 22 2.42 130

1.78 27 1.45 14 2.19 70

α ¼ 90 α ¼ 315

β MRI β MRI β MRI

current directionality factor approach is more significant. The only approximation involved in the proposed approach is the neglect of influence of correlation of directional wind speeds. As discussed previously, this influence is negligible in general.

under directional design wind speeds, which is similar to the sector-by-sector approach. However, instead of using R-year directional wind speeds, the directional wind speeds are determined considering the dependence of response character on direction, referred as directional equivalent wind speed, which are determined as follows. The R-year response xR is firstly quantified using a comprehensive approach developed by Matsui and Tamura (2004), which considers the directionality effect of wind speed and response. The directional wind speed in ith wind pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2xR =ρci , direction causing response level xR is calculated by vixR ¼ where ci is the extreme response coefficient. The corresponding MRI of vixR is then calculated. The worst wind direction with the lowest MRI, say, R’ years, where R’  R, among all directions is determined, and the wind speed in this direction is recorded. R’ is referred to as equivalent MRI for this R-year response xR . The above calculation is repeated for several responses of interest and for different structural orientations. The average of these recorded directional wind speeds in each direction is defined as the design directional wind speed in this direction. Table 3 gives the equivalent MRI (TE ) of wind speed for 50-year internal force of S2. For S2 under Taps 87, 86 and 85 at α ¼ 90 degrees, there is only one dominant direction, NW direction, the equivalent MRI is identical to the target one, i.e., 50 years. At α ¼ 0 and 315 , there are more than one dominant direction, the equivalent MRI is greater than 50 years. Here, 4 internal forces of screws (S2 under Taps 87, 86, 85; S2 under Taps 3, 21, 39; S2 under Taps 16, 34, 52; S2 under Taps 253, 254) under 8 structural orientations are considered, thus there are 4 8 ¼ 32 cases. Table 3 only shows some results for brevity. The equivalent wind speed (VE ) for each wind response associated with one wind direction is picked out. For example of S2 under Taps 87, 86 and 85, the smallest TE ¼ 89 year at NW direction with VE ¼ 24.6 m/s is selected. The selected 32 responses leads to 32 wind speed data distributed among eight wind directions. The average of wind speeds in each wind direction gives the equivalent wind speed V E . As VE is different from V E , some deviations may be caused by use of V E for calculating some specific wind effects. The deviations for the considered 32 internal forces are about from 14.4% to 11%. From above discussions, it is clear that use of directional wind speeds as adopted in AS/NZS 1170.2:2011 and AIJ-RLB-2004 represents an improved definition of directionality effect as compared to single directionality factor used in ASCE 7-10. Here, a further improvement of this type of approaches is proposed. Considering the fact that the influence of correlation of directional wind speeds is insignificant in general, the directional wind speeds can be modeled as independent. Therefore, Eq. (4) or Eqs. (5) and (6) can be used to estimate the yearly maximum distribution of response with consideration of directionality, from which the response for a given MRI can be subsequently determined. Similar to the approach in AS/NZS 1170.2:2011, the definition of directional wind speed is not affected by directional response characteristics. This newly proposed approach is more accurate than those used in AS/NZS 1170.2:2011 and AIJ-RLB-2004. For the 32 cases addressed in this study, the deviation is less than 5%. The advantage of this approach over the

3.4. Estimation of screw internal forces with various MRIs based on their JCDF 3.4.1. JCDF of multiple screw internal forces with consideration of directionality Based on the extreme wind speed and extreme response coefficients information in section 3.1 and 3.2, JCDF of multiple screw internal forces with consideration of directionality can be estimated by Eq. (8). Fig. 16 plots the contour of JCDF of internal forces of screws S2 and S3 (under Taps 87, 86 and 85) when α ¼ 0 , denoted as “Real-Eq. (8)”. The cases with full correlation and independent assumptions of response coefficients at same wind direction by Eqs. (9) and (10), denoted as “Full correlation, directional-Eq. (9)” and “Independent, directional-Eq. (10)”, are also shown. The 50-year (i.e., CDF is 0.98) internal force of S2 and S3 estimated by their separate CDFs, i.e., Eq. (3), are ws2 ¼ 575 N and ws3 ¼ 562 N. The JCDF at these response levels are 0.9762, corresponding to MRI of 42 years. Under full correlation and independent assumptions, the JCDFs are 0.9780 and 0.9743, with MRIs of 45 and 39 years, respectively. Similarly, the 500-year (i.e., CDF is 0.998) internal forces estimated by their separate CDFs are 916 N and 899 N. The JCDFs are 0.99755, 0.99774 and 0.99713, with MRIs of 408, 442 and 348 years, under partial correlation, full correlation and independent cases. Furthermore, the JCDF may be calculated directly from the separate CDFs of internal forces under the assumptions of full correlation and independent by Eqs. (13) and (14), denoted as “Full correlation-Eq. (13)” and “Independent-Eq. (14)” shown in Fig. 16. In these cases, the JCDFs at 50-year response levels, i.e., ws2 ¼ 575 N and ws3 ¼ 562 N, are 0.98 and 0.96, with MRIs of 50 and 25 years, respectively. Similarly, the JCDFs at 500-year response levels are 0.998 and 0.996, with MRIs of 500 and 250 years. It should be mentioned that the independent case should be estimated by “Independent, directional-Eq. (10)”, not by “Independent-Eq. (14)”. Fig. 17 shows another case. The 50-year and 500-year responses for S2 and S3 by their separate CDFs are 297N and 332N, 457N and 524N. The JCDFs and MRIs of these response levels are 0.96695 and 0.99576 with MRIs of 30 and 235 years, in "Full correlation, directional-Eq. (9)" case. In "Independent, directional-Eq. (10)" case, these are 0.99557 and 0.99545 with MRIs of 28 and 220 years. Obviously, the correlation between these two responses at same wind direction has less influence in this particular case. 3.4.2. Estimation of system reliability In this section, the resistance is incorporated to estimate the system reliability. The mean of uplift strength of screw fastener of trapezoidal

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acceleration, building story drift and bending moments.

steel cladding with closely spaced ribs corresponding to local dimpling or pull-through failure can be calculated as (Mahaarachchi and Mahendran, 2009): 

μR ¼ 0:008 9:52 

49fy d Et

2  3=4  1=5  1=3 hc wt 1500t 2 dtfy 12 þ p wc Ld

Acknowledgments The supports provided in part by National Natural Science Foundation of China (51808077, 51720105005, 51478401), 111 Project of China (B18062), China Postdoctoral Science Foundation (2017M622966), Chongqing Postdoctoral Science Foundation (XmT2018039), and Chinese Fundamental Research Funds for the Central Universities (2018CDXYTM0003) are greatly acknowledged. The authors also thank Professor Yukio Tamura and Professor Masahiro Matsui for their help and valuable discussions on directionality effect.

(22)

where the steel yield stress fy is 250 Mpa for G250 grade steel; the diameter of screw head d is assumed to be 11 mm; the Young’s modulus E is 200 Gpa; the metal thickness of cladding t of 0.6 mm is used; the crest height hc , crest width wc , trough width wt and pitch p as shown in Fig. 7(b) are 30 mm, 43.5 mm, 81.5 mm and 125 mm; the span L between purlins is 2000 mm. Considering the variations in material, fabrication and loading effects, a capacity reduction factor ε ¼ 0.63 for resistance is suggested (Mahaarachchi and Mahendran, 2009). By using the above selected parameters, μR is 586 N. The COV is assumed to be 0.1. The resistance is assumed to follow the lognormal distribution. The system reliability associated with single and multiple limit state responses is further estimated by the AS/NZS, AIJ, ASCE standard and proposed approach, where the multiple limit state responses are assumed to be independent. The results are compared to that by the multivariate model as shown in Table 4, which correspond to the cases in Table 3. It is evident that AS/NZS 1170.2:2011 approach underestimates the risk, while AIJ-RLB-2004 and ASCE 7-10 approaches can lead to over- or underestimation of risk. The proposed method gives the best estimation which will not underestimate the risk. The system reliability with multiple limit state responses depends on the sum of failure probability of each limit state response under independent assumption. Thus, the performance of AS/NZS 1170.2:2011, AIJ-RLB-2004, ASCE and proposed method for system reliability of multiple limit state responses will be similar to that of single limit state response. Table 5 gives the reliability index β and estimated MRI of single screw and system with multiple screws under Taps 87, 86 and 85 with different structural orientations when the response coefficients are treated as random variables. The internal force of S2 would become larger when considering the uncertainty of response coefficient (Chen and Huang, 2010; Zhang and Chen, 2015), so the MRI of S2 in Table 5 decreases compared to that in Table 4. The system reliability index and MRI are lower than that of each single screw. The risk is overestimated when the responses of screw S2 and S3 regardless of direction are assumed to be mutually independent.

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4. Conclusions The directionality factor can vary in a wide range for different responses with different structural orientations. Instead of introducing single directionality factor for all responses, the revised sector-by-sector approach adopted in AIJ recommendation represents much improvement in modeling directionality effect. The AIJ approach can result in different directionality factors for different responses via using the same procedure in terms of design directional wind speeds, while some responses can be under- or overestimated. The alternative approach proposed in this study considers the directionality effect under the independent assumption, which is more accurate and relatively easy for practical application. The new framework introduced in this study leads to estimation of JCDF of multiple limit state responses with consideration of directionality effect. The system reliability was then calculated by further modeling the uncertainties of resistances. The correlation of directional wind speeds and correlation of multiple limit state responses at same wind speed and direction can have important influence on system reliability. Examples and parameter studies were presented using responses of claddings on a large-span saddle type roof. The proposed framework can also be used for different kinds of wind responses, such as maximum

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Simiu, E., Filliben, J.J., 2005. Wind tunnel testing and the sector-by-sector approach to wind directionality effects. J. Struct. Eng. 131 (7), 1143–1145. Warsido, W.P., Bitsuamlak, G.T., 2015. Synthesis of wind tunnel and climatological data for estimating design wind effects: a copula based approach. Struct. Saf. 57, 8–17. Wen, Y., 1983. Wind direction and structural reliability. J. Struct. Eng. 109 (4), 1028–1041. Wen, Y., 1984. Wind direction and structural reliability: II. J. Struct. Eng. 110 (6), 1253–1264.

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