Engineering Structures 27 (2005) 89–95

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Reliability-based assessment of roofs in Japan subjected to extremsnows: incorporation of site-specific data

Toru Takahashia, Bruce R.Ellingwoodb,∗

aChiba University, JapanbGeorgia Institute of Technology, School of Civil and Environmental Engineering, 30332 0355 Atlanta, GA, USA

Received 18 March 2003; received in revised form 24 August 2004; accepted 7 September 2004

Available online 26 October 2004

Abstract

Performance-based engineering is a new paradigm in structural design for natural hazards. In this paper, the authors explore the problemof estimating regional variations in fractiles of snow load used for design purposes from a limited number of discrete observing stationreliability implications of such estimates are examined forthe recent (2000) revision of theJapanese Building Code (JBC), which introducedconcepts of performance-based design, and permitted the estimation of ground snow load for determining site-specific snow load data. Akriging technique was used for estimating the spatial distribution of snow weight on the ground from a finite array of weather stations aground snow is recorded. Special treatment of snow data may be required for mountainous areas. The result ofa reliability-based analysis ofthe new code provisions indicated that steel roof members designed by theJapanese Building Code appearto have a lower reliability againstsnow load compared to roof members designed by codes in the United States, Canada or Western Europe. The reasons for this diffeexplored and future research needs are identified.© 2004 Elsevier Ltd. All rights reserved.

Keywords: Buildings (codes); Design (buildings); Limit states; Loads; Probability; Reliability; Snow; Statistics; Structural engineering

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1. Introduction

The estimation of regional variations in environmenloads is a perennial problem in structural code developmebecause the weather stations where high-fidelity datarecorded often are widely separated. Thus, the observalues seldom can be used directly in site-specific desidirectly, and some interpolation method is requireddetermine appropriate local design loads [1]. Systematictechniques are required to develop snow maps from suobserving stations for code purposes. As a case in pothe Japanese Building Code (JBC) was revised in 2000,and concepts of performance-based design were introduced.In performance-based design, environmental loads musestimated at different mean recurrence intervals (MRI)achieve suitable performance goals for different catego

∗ Corresponding author. Tel.: +1 404 894 1635; fax: +1 404 894 2278.E-mail address: bruce.ellingwood@ce.gatech.edu (B.R. Ellingwood).

0141-0296/$ - see front matter © 2004 Elsevier Ltd. All rights reserved.doi:10.1016/j.engstruct.2004.09.001

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of building occupancy. The newJBC allows the structuralengineer to perform a site-specific analysis of snow load apart of the structural design process.

One particular difficulty with theJBC specification ofsnow load has been noted [2]. In the JBC, the groundsnow loads at specific building sites are determined froma regression formula that is a function of the altitude ofthe site and sea/land ratio measured with respect to 20and 40 km radii centered on the site. TheJBC providesa table defining the regression coefficients for calculatinga ground snow depth with a 50-year mean recurrencinterval (MRI) at each site. In some areas, the residuafrom this estimation procedure are quite large. But theJBC provides no additional information that is usableby a designer on the residuals at each site associatwith this estimation procedure, leaving questions regardinthis source of uncertaintyand confidence in the site-specific snow load estimates untouched. In this paperwe evaluate the information contained in these residuals

90 T. Takahashi, B.R. Ellingwood / Engineering Structures 27 (2005) 89–95

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Fig. 1. Parameters defining a spherical variogram.

and their contribution to the variance in the design snoload, and assess their impact on the first-order (Freliability [ 3] of simply supported roof members designeby the JBC. These evaluations lead to recommendationfor code improvements and additional supporting researWhile the illustrations focus on engineering practiceJapan, the general methodology and issues are germanother contexts where regional variation on environmenloads is of concern.

2. Spatial data estimation

2.1. Kriging

There are several techniques for spatial estimationdata, including multiple regression analysis, spline intepolation, nearest-neighbor interpolation, moving averagand kriging. In this paper, the authors used a kriging tecnique for interpolating between weather stations. Kriginga method for estimating the values of a parameter at a pticular site using observed data taken at nearby sites [4,5].In kriging, a variogram (or semi-variogram) is determinewhich is a function of distance between observation pointsand describes the spatial coherence (or stochastic depdence) among the data. Such variograms provide a secomoment depiction of the ground snow, modeled as a spatiadistributed stochastic field. There are numerous forms of viogram, including linear, power, spherical, exponential, aGaussian, that can bechosen to model the spatial coherencin the data, as described in the following.

2.1.1. Co-variogram and semi-variogramA co-variogram requires second-moment stationarity

the field. The co-variogramC(d) for random field Z isexpressed as follows:

C(d) = 1

|N(d)|∑N(d)

[Z(si ) − µZ ][Z(s j ) − µZ ] (1)

.

o

,

n--

whered = distance between sitessi ands j , N = numberof pairs at distanced, and Z = value of the field at pointssi or s j . In contrast, a semi-variogram (or, for simplicity,a variogram) γ (d) requires intrinsic stationarity, and isexpressed as follows:

γ (d) = 1

2|N(d)|∑N(d)

[Z(si ) − Z(s j )]2. (2)

Second-moment stationarity is not guaranteed for grousnow loads modeled as a random field. Therefore, tauthors adopted the semi-variogram (Eq. (2)) in this study.

2.1.2. Snow load model variogramThe spherical variogram is widely used in krigin

analysis because of its flexibility. It is expressed as follow

γ (0) = θ1 ‖d‖ = 0

γ (d) = θ1 + θ2

⌊3‖d‖2θ3

− 1

2

(‖d‖θ3

)3⌋

0 < ‖d‖ ≤ θ3

γ (d) = θ1 + θ2 θ3 < ‖d‖.(3)

The parameters in Eq. (3) are interpreted in Fig. 1. Thenugget effect (defined by parameterθ1) describes samplingerror or short-scale variability, therange (parameterθ3)defines the region over whichthe effect of distance appearin the variogram, and thesill (sumθ1 + θ2) places a limiton the maximum value of the variogram outside of the ranlimit distance.

2.2. Estimation of ground snow loads by kriging

In Hokkaido, Japan, there are 161 meteorologicstations that automatically collect data on precipitatiand temperature. The AMeDAS (Automated MeteorologicData Acquisition system) began operation in 1976, andtypical record length (through the year 2000) is 24 yeaThese data have been published by the Japan MeteorologAgency [6]. The ground snow weight for an MRI of 100years for each station was estimated using a snow lamodel [7,8]. For the kriging analysis, theS+SpatialStats [9]analysis package was used. The nugget(θ1), sill (θ1 + θ2)

and range(θ3) parameters for Hokkaido were found to be10, and 70, respectively. The empirical and resulting modvariograms developed using these parameters are compin Fig. 2. The estimated weight of snow on the ground fa 100-year MRI and the standard error of this estimateshown by thecontours inFigs. 3and 4, respectively. Thedots represent the sites of the 161 meteorological stationHokkaido.

The map thus determined is suitable for ordinary (nomountainous) sites, but may not be acceptable formountainous regions or other regions with very heavy snaccumulation. There are only a few stations where dare collected in mountainous areas, and local variationsterrain and elevation can be extreme. Thus, when usthe estimated snow weights for the design of structu

T. Takahashi, B.R. Ellingwood / Engineering Structures 27 (2005) 89–95 91

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Fig. 2. Empirical and analytical variograms for ground snow in Hokkaido,Japan.

Fig. 3. Estimated 100-year ground snow load in Hokkaido (kN/m2).

Fig. 4. Standard error in estimated snow load in Hokkaido (kN/m2).

in mountainous areas, special considerations are neeAs shown in Fig. 4, the residuals estimated from thekriging procedure often are not especially large, evin mountainous areas (e.g., just north of Cape Erimbecause the estimates from the kriging analysis do not

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Fig. 5. Structural model of roof member used in FO reliability analysis.

reflect local topographical variations in such areas. Instethey are calculated simply as a function only of distanbetween observation points, as expressed in Eq. (2). Thetrue residuals in ground snow load in mountainous areestimated by ordinary kriging cannot be predicted withoadditional meteorological stations. However, the additionweather stations is costly and fewer buildings are expecto be built in mountainous areas than in flat areas. Therefosome tradeoffs must be made between the cost of addiweather stations vs the cost of designing with snow loadthat reflect the higher residuals in the estimates of the desibasis snow load (e.g., MRI of 50, 10 or 500 years, dependon the building performance objective).

In the following section, the authors illustrate thimportance of this point by examining how the reliabilityof a roof member depends on the variance in the snow loestimate.

3. Structural reliability of roofs

The reliability of a roof structure depends on the varianof both loads and resistance. The reliability of simplysupported roof members, as shown inFig. 5, wasassessedusing first-order (FO) reliability analysis [3]. The limit stateis defined by formation of the first plastic hinge; it isassumed that roof purlins provide lateral support, and neitlateral–torsional buckling nor shear failure are considereInasmuch as the 2000 edition of theJBC permits bothAllowable Stress Design (ASD) and Limit States Desig(LSD), both approaches are considered.

3.1. Resistance in allowable stress design (ASD)

The flexural resistanceR of the simply supported roofbeam designed by the allowable stress design (ASD) methis:

R = B(Fy/Fyn)(Zx/Zn) f R1 (4)

where B = professional factor [10], Fy = yield strength,Fyn = nominal yield strength,Zx = plastic section modu-lus, Zn = nominal value of plastic section modulus,f =shape factor, andR1 is determined by the specific combination of design loads, as indicated in Eqs. (5a) and (5b) below.The basic parameters used in Eq. (4) arelisted inTable 1.

Specific load combinations for allowable stress desiare listed in the 2000 edition of theJBC. The requiredroof beam for snow load is determined from the maximuvalue of these combinations, as shown in Eqs. (5a) and (5b).

92 T. Takahashi, B.R. Ellingwood / Engineering Structures 27 (2005) 89–95

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Table 1Parameters used in Eq. (4) [10]

Variable Mean COV CDF

Steel H shape beamB 1.02 0.06 NormalFy/Fyn 1.05 0.11 LognormalZx/Zxn 1.01 0.05 Normalf 1.12 n/a n/aFs 1.5 for long term loading(FS1),

and 1.0 for short term loading(FS2)

For ordinary regions:

R1 = max

{(Dn + Ln)FS1

(Dn + Ln + Sn)FS2

}. (5a)

For snowyregions:

R1 = max

(Dn + Ln)FS1

(Dn + Ln + 0.7Sn)FS1

(Dn + Ln + Sn)FS2

(5b)

where Dn = nominal value of dead load,Ln = nominalvalue of live load,Sn = nominal value of snow load,FS1 =safety factor for long term loading, andFS2 = safetyfactorfor short term loading. The factorsFS1 and FS2 and thefactor 0.7 in Eq. (5b) were introduced intheJBC many yearsago to address the impact of snow load duration on creof wood and reinforced concrete, and were carried ovto other structural materials by engineering judgment [11].For ordinary regions, when the value ofSn/Dn is low, therequired strength is governed by long term loading, but fa large span steel structure, i.e. whenSn/Dn is large, therequired strength is defined by short term loading with snoload. For snowy regions, the equation(Dn + Ln +0.7Sn)FS1

always leads to the largest required strength, and therefthe resistance is defined by long term loading.

3.2. Resistance in limit state design (LSD)

The flexural resistanceR of the simply supported roofbeam using LSD is defined by

R = B(Fy/Fyn)(Zx/Zn)(γD Dn + γL Ln + γS Sn)/φR . (6)

The load combination requirements for limit states desiare listed in the 2000 edition of theJBC. For ordinaryregions:

Dn + Ln + 1.4Sn

Dn + Ln + 1.6Wn

Dn + Ln + 0.35Sn + 1.6Wn

(7a)

while for snowy regions:

Dn + Ln + 1.4Sn

Dn + Ln + 1.6Wn,(7b)

The load factors for snow and wind loads are based onratio of their 500-year MRI to 50-year MRI values.

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3.3. Dead load

Dead load (normalized by the nominal value of deaload), D/Dn , is defined by a normal distribution, with ameanµD = 1.05 and coefficient of variationVD = 0.10.

3.4. Roof live load

It is common to use 60 kg/m2 (0.59 kN/m2) for rooflive loads in Tokyo. Therefore, the effect of this load wasincluded in the analysis.

3.5. Snow load

Snow loadS [N/m2] on the roof is defined in theJBC as

S = dρsCS gCe (8)

whered = snow depth [m], ρs = snow density [kg/m3],CS = ground-to-roof conversion factor,g = gravityacceleration [m/s2], andCe = environmental factor.

3.5.1. Snow depthThe annual maximum snow depth is described by

Gumbel distribution. Following the Architectural Institute oJapan (AIJ) recommendations for loads on buildings [12],the ratio of the mean value of the 50-year maximaS tonominalvalue Sn , i.e. the value with an MRI of 50 years,was assumed from thefollowing information:

d = 1.11dn (d100 < 1 m) (9a)

d = 1.08dn (d100 ≥ 1 m) (9b)

whered100 is the snow depth for an MRI of 100 years. TheCOV of the annual maximum snow depth varies station bstation, as indicated inFigs. 3 and 4. In this paper, threedistinct values of coefficient of variation, i.e., COV equal to0.07, 0.2, and 0.4, for snow depths were considered.

3.5.2. Snow densityThe minimum unit snow weight is defined in theJBC

as 2 kN/m3, and 3 kN/m3 is commonly used for designin snowy regions. On the other hand, the following snowdensity is recommended based on in situ measurements [13].

ρS = 73√

d/d0 + 240 [kg/m3] (10)

whered0 is a normalizing snow depth (1 m), introduced sothat parameterd/d0 within the square rootis dimensionless.In this paper, it was assumed that Eq. (10) defines the actualsnow density.

3.5.3. Ground-to-roof conversion factorIn the JBC, the following equation is used to convert

ground snow load to roof snow load. It follows the simplesequation inISO Standard 4355 [14].

CS = √cos(1.5θ) (11)

T. Takahashi, B.R. Ellingwood / Engineering Structures 27 (2005) 89–95 93

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whereθ = roof slope (in degrees). Of course, the actuaconversion factor varies from case to case. Although theresome data on the ground-to-roof conversion factor [15–19],the authors assumed for flat roofs thatCS is described bya normal distribution, with mean valueµCS = 0.7 andcoefficient of variationVCS = 0.4 [15]. It is recommended intheJBC that unevenly distributedsnow load onroofs due todrifting or aerodynamic shade should be treated by designeon an individual basis. Since the roof geometries considerherein were relatively simple, the effect of unbalanced loawas neglected in this study.

3.5.4. Environmental factorIn the AIJ recommendations, the environmental factor a

counts for meso-scale exposure (topography and surrouning largeobjects and obstructions) and depends on the builing site. This effect was neglected in this paper.

3.6. Limit state function

The limit state functionG is defined as follows:

G = R − (D + S) (12)

where theregion G < 0 means failure of the structure.In the analysis, Eq. (12) was normalized byDn , and theratio of nominal snow to dead load,Sn/Dn , was used as theparameter to illustrate the sensitivity of the roof structurereliability to roof system and snow climatology.

G = R

Dn−

[D

Dn+

(S

Sn

) (Sn

Dn

)]. (13)

In areas where snow is a significant design consideratiothe ratio Sn/Dn can vary from approximately 1 to 10(or more) in mountainous regions. ParameterR/Dn wasmodeled by a lognormal distribution,D/Dn was modeledby a normal distribution, andS/Sn was modeled by alognormal distribution [10]. In first-order reliability analysis,the metric of reliability is the reliability index,β, defined asthe minimum distance from the failure surface to the origiin a space of reduced or unit random variables obtaineby subtracting the mean and dividing by the standardeviation of the basic variables in the analysis [3], here theflexural strength, dead, live and snow loads, respectiveThe service life of the structure was assumed to be 50 yeathus, the load statistics were derived for 50 years, and treliability index is based on this reference period. For readeaccustomed to think in terms of a probability of failureP f (again, for a reference period of years), it is relateapproximately to the reliability index byPf = Φ(−β).

3.7. Results of reliability analysis

The results of the first-order reliability analysis of steeroof members are shown inFigs. 6a–6c, showing reliabilityindices for framing members in steel roof structures forthree different ground snow loads, representing three distin

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Fig. 6a. Reliability indices for members designed by ASD and LRFDordinary snow region(dn = 0.27 m).

Fig. 6b. Reliability indices for members designed by ASD and LRFDordinary snow region(dn = 0.8 m).

Fig. 6c. Reliability indices for members designed by ASD and LRFDsnowy regions(dn = 2.0 m).

climates. In ordinary regions, since the allowable stressequal to the yield stress for short term loading, the values

94 T. Takahashi, B.R. Ellingwood / Engineering Structures 27 (2005) 89–95

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reliability indices are almost constant forSn/Dn greater than1.0. The difference betweenFigs. 6aand6bis caused by thedifference between unit snow weight used in the design aactual snow density assumed in Eq. (10) andby the effectof the basic live load considered in the design. In a lighsnow region, such as Tokyo, the requirement to consider liload on the roof increases the reliability of the roof membagainst snow load as well.

On the other hand, in snowy regions the parameterR1[Eq. (5)] associated with long term loading invariably hathe largest value. Therefore, reliability indices for ASare relatively high whenSn/Dn is low, and decrease whenSn/Dn becomes larger. In every case, the reliability indicfor LRFD are nearly constant asSn/Dn varies, as mightbe expected with a different load factor on dead and snload [20]. However, the values of the reliability index arlower than those associated with LRFD or LSD criteriathe United States, Canada orWestern Europe, where theβsare on the order of 3 on a 50-year basis or 4 on an annbasis [21,22].

Fig. 6 shows that if the variance in the snow load issmall, the reliability is high. However, there are two kindof uncertainty that contribute to the overall variance isnow load, and the overall variance in snow load cannoteliminated completely. The first is “inherent randomneswhich is generally irreducible at the current state of the aThe second is “knowledge or modeling uncertainty,” whicmay reducible with some additional cost of data collectioand analysis. As indicated inFigs. 3 and 4, the snowload statistics and distributions at points between weathestations are uncertain, even if the values of the residualstandard error are small. This source of uncertainty woube reduced (although not eliminated entirely) if additionalweather stations were to be placed in these areas. Tcourse of action might be very costly. Moreover, even if textreme values for design, i.e. snow depth or snow weigon the ground, could be estimated accurately, uncertain the conversion factor and the other factors generacauses the COV of snow load to be large. Therefore, furtinvestigations aimed at characterizingonly snow depth orsnow weight on the ground may not reduce the variancesnow load significantly.

Fig. 7 compares the reliability indices of roof structuramembers for ASD and LRFDin the JBC, LRFD in theAIJ [23], and LRFD in AISC and ASCE Standard 7 [21].The load and resistance factors used in the analysis are liin Table 2. The loadand resistance factors listed in theJBCappear to be unconservative for designing roofs against sload when compared to other international recommendationand standards. Although theJBC is the minimum designrequirement, the code values are commonly used in roustructural design of roofs. Since snow load is one of thgravity loads, once a collapse mechanism initiates, tstructure as awhole may become unstable. Accordingly,additional consideration regarding the reliability level fosnow load in theJBC appears warranted.

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Fig. 7. Reliability indices for members designed by ASD and LRFD isnowy regions using JBC, AIJ and ASCE codes (dn = 2.0 m, V s = 0.07).

Table 2Loads and resistance factors used in the analysis shown inFig. 7

γD γL γS φ

LRFD in JBC 1.0 1.0 1.4 1.0LRFD in AIJ, β = 2.0 1.02 0.46 1.23 0.83LRFD in AIJ, β = 3.0 1.03 0.47 1.56 0.77LRFD in ASCE 1.2 0 1.6 0.9

4. Concluding remarks

Snow load varies from region to region in Japan. Thsnow loads forpractical design have been designateby each local prefectural government and the criteriofor determining snow load had been varied similarlyFurthermore, it has been the custom to remove snow froroofs in snowy regions, and accordingly the nominal snoloads have been designated at relatively low levels. In 199the AIJ recommended that all loads on buildings be set at anMRI of 100 years for use with standards based on allowabstress design. In 2000, the JBC was changed and loadsan MRI of 50 years were designated for allowable stredesign for snow and wind load. [In contrast, earthquake lois based on an MRI of 500 years, with a load factor equal to1.0.] As shown in this paper, the resulting criterion for designsnow load in Japan might not be sufficiently conservativeon three accounts. First, the safety factor for snow loais not conservative against snow load acting as a gravload. Second, the unit snow weight (snow density) is nconservative especially for nominal snow depth betweenand 100 cm. Third and finally, the load and resistance factofor limit state design are also not conservative because thvaluesare calibrated with existing buildings, i.e. existinglocal codes, which had been developed under the assumpthat heavy snows would be removed from roofs, as habeen historical custom. Thus, the reliability of building roofagainst snow load in Japan may be unconservative whcompared with those designed by codes in the United Stat

T. Takahashi, B.R. Ellingwood / Engineering Structures 27 (2005) 89–95 95

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Canada and Western Europe. These apparent differencesstructural reliability provided by national design standardraise points for discussion in the international standarwriting bodies such as ISO and create a safety issue on whfurther research would be desirable.

As a final observation, the map for snow load in themountainous areas of Japan (e.g.,Figs. 3 and 4) is basedon limited climatological data. A number of importantfactors, including the dependence of snow load on prevailintemperature and elevation [24], seldom are reflected insuch maps. One would draw similar conclusions from anexamination of weather station networks in mountainouregions in Western Europe, Canada and the United StatThis lack of technical support for snow load design criterihas important implications for performance and reliabilityof buildings subjected to extreme snow loads in such areaand should receive further attention from the internationalstructural standards community.

References

[1] Tallin AG, Ellingwood BR. Structural load estimates fromgeographically sparse data. J Struct Eng, ASCE 1987;113(3):628–32.

[2] Mihashi H, Takahashi T, Okada H. Recent revision of Japanesbuilding code on snow load and further research needs. J Snow EJapan 2001;17(2):95–105.

[3] Melchers RE. Structural reliability: analysis and prediction. JohWiley; 1997.

[4] Olea RA. Geostatistics for engineers and earth scientists. KluwerAcademic; 1999.

[5] http://www.ems-i.com/gmshelp/interpolation/interpolation_schemeskriging/kriging.htm.

[6] Japan MeteorologicalAgency: Surface Daily Processed Data (CD-ROM), 1961–1999, Automated Meteorological Data AcquisitionSystem Data (CD-ROM), 1976–1999, Japan Meteorological Agency

[7] Takahashi T, Kawamura T, Kuramoto K. Estimation of ground snowload using snow layer model. J Struct Construct Eng (TransactionsAIJ) 2001;(545):35–41.

n

-h

s.

,

f

[8] Izumi M, Mihashi H, Takahashi T.Statistical properties of the annualmaximum series and a new approach to estimate the extreme valuefor long return periods. In: Proc. of 1ICSE, CRREL Special repo89-6, 1989. p. 25–34.

[9] http://www.msi.co.jp/splus/tips/spatial/frame.html.[10] Galambos TV, Ellingwood BR, MacGregor JG, Cornell CA

Probability-based load criteria: assessment of current design practJ Struct Div, ASCE 1982;108(5):959–77.

[11] Ohashi Y. The history of structural codes of buildings in Japan. JapBuilding Center; 1993.

[12] AIJ. Recommendations for loads on buildings. Architectural Instituof Japan; 1993.

[13] Joh O, Sakurai S. Equivalent snow density in heavy snowing arJ Snow Eng, Japan Society for Snow Engineering (JSSE) 1993;9(112–114 [in Japanese].

[14] ISO. ISO4355 bases for design ofstructures—determination of snowloads on roofs. 1998.

[15] O’Rourke MJ, Redfield R. Uniform snow loads on structures. J StruDiv, Proc. of ASCE 1982;108(ST12):2781–98.

[16] Taylor DA. Roof snow loads in Canada. Canadian J Civil Eng 1987(1):1–18.

[17] Mihashi H, Takahashi T, Izumi M. Wind effects on snow loads. InProc. of 1ICSE, CRREL Special report 89-6, 1989. p. 158–67.

[18] Mihashi H, Takahashi T, Izumi M. Field measurement andcharacteristic analyses of snow load on flat roofs. Proc. of 2ICSCRREL Special report 92-27, 1993. p. 69–80.

[19] Tomabechi T, Yamaguchi H, Hashimoto S. An evaluating methoof snow depth on roofs in relation with meteorological conditionsJ Snow Eng Japan 1993;9(3):206–13.

[20] Ellingwood BR et al. Probability-based load criteria: load factors anload combinations. J Struct Div, ASCE 1982;108(5):978–97.

[21] ASCE Standard 7-02. Minimum design loads for buildings andother structures. Reston (VA): American Society of Civil Engineers;2002.

[22] CEN 250. Eurocode 1—actions on structures. Brussels, BelgiuEuropean Presta-1, Comite Europeen de Normalization 251994.

[23] AIJ committee of limit state design: Tentative recommendations flimit state design of buildings, Ver. 2. Architectural Institute of Japa2000.

[24] Takahashi T, Shitara T, Ellingwood BR. Ground snow load estimatiowith Kriging. In: Proc., 5th Int. conf. on snow engineering, (Paper SE04-061). Rotterdam: AA Balkema; 2004. p. 135–40.