Background Report EN 13031-1: 2019, Annex C

Snow Loads on Greenhouse Roofs Part II-2: Snow load distribution

Multi-pitched roofs

Photo VDH: Melting down 65 cm snow per 8 h on a greenhouse roof

Autors: Dr.-Ing. I. Pertermann, IB Puthli, Schüttorf Prof. Dr.-Ing. R. Puthli, KIT, Karlsruhe Date: 22.02.2021

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Table of content Symbols ………………………………………………………………………...…… 3 1. Basic principles of EN 1991-1-3 …………………………………………………. 6

Part I: Melting

Part II-0: Sliding and Drift General Background and Classification of Greenhouses

Main document

Part II-1: Snow load distributions Duo-pitched roofs

Part II-2: Snow load distributions

Multi-pitched roofs 9. Multi-span pitched greenhouse roofs in EN 13031-1: 2019 ………………..…. 8

9.1 General requirements for greenhouses regulations ……………………………….… 8

9.1.1 Thermal coefficient Ct ………………………………………………….…… 8

9.1.2 Surface material coefficient Cm …………………………………………… 11

9.1.3 Classification of greenhouse roofs in EN 13031-1: 2019 …………………. 13

9.2 Measurements on multi-span roofs …………………………………………………. 14

9.3 Snow load distributions - models in different standards ………………………..… 17

9.3.1 Multi-span roofs in the Eurocode - present and future ………………..…17

9.3.2 Complete redistribution model in ISO 4355: 2013 …………………….… 21

9.3.3 Incomplete sliding model in ASCE 7-10 ………………………………..… 24

9.4 Conclusions and greenhouse model in EN 13031-1: 2019 ………………………..…26

9.5 General redistribution model for multi-span roofs – Proposal …………………… 30

Literature to Part II-2 …………………………………………………………..… 35

Note: Background details to 9.1 can be found in Part I and Part II-0 of this Background report. This is a corrected version. In chapter 9.5 the drift losses above the trough are considered within

the model itself, not any more as an additional correction.

Part II-3: Snow load distributions Arched roofs

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Symbols Abbreviations:

Luv windwardsideofthestructure

Lee leewardsideofthestructure

Symbols:

NOTE ThefollowingsymbolsarebasedonEN1990andEN1991.

Latinupper-caseletters:

Ce exposurecoefficient

Cesl exceptionalsnowloadcoefficient

Cm surfacematerialcoefficient

Ct thermalcoefficient

N numberofdayswithvm,10>5m/sduringthethreecoldestmonthsoftheyear

Rsi internalsurfaceheattransferresistanceinm2K/W:Rsi=1/hi

Rse externalsurfaceheattransferresistanceinm2K/W:Rse=1/he

Sr,1 totalroofsnowloadinkNforloadcase1

Sr,2 totalroofsnowloadinkNforloadcase2

U overallheattransmittanceinW/(m2K)

Uo specialheattransmittanceforsnowmeltconditionsinW/(m2K):Uo=U/(1–U·Rse)

W widthofarooftroughofamulti-pitchedroofinm

Latinlower-caseletters:

d depthofthesnowinm

fd(Ce) driftsurchargefactorforpitchedroofs

fd(x;Ce) driftsurchargefactorforarchedroofs

hd roofheightinm;archedroofISO-format

hr roofheightinm

he externalsurfaceheattransfercoefficientinW/(m2K),total:he=hr,e+hc,e

hr,e externalsurfaceheattransfercoefficientforradiationinW/(m2K)

hc,e externalsurfaceheattransfercoefficientforconvectioninW/(m2K)

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hro externalsurfaceheattransfercoefficientforBlack-Body-radiationinW/(m2K)

hi internalsurfaceheattransfercoefficientinW/(m2K),total:hi=hr,i+hc,i+hcd,i

hr,i internalsurfaceheattransfercoefficientforradiationinW/(m2K)

hc,i internalsurfaceheattransfercoefficientforconvectioninW/(m2K)

l roofwidthorspaninm

l1 snowcoveredroofwidthinm

n returnperiodinyears

s roofspaninm

s snowloadinkN/m2

sb basicroofsnowloadinkN/m2ISO-format

sd driftsnowloadinkN/m2ISO-format

ss slideloadinkN/m2ISO-format;slideloadlimitinkN/m2

sgm roofsnowweightattheslidelimitinkg/m3

s0 characteristicgroundsnowloadinkN/m2ISO-format

sk,n characteristicgroundsnowloadinkN/m2forthereferenceperiodn

si,n,t characteristicroofsnowloadinkN/m2forthelocationi,theshapecoefficientµi,thereferenceperiodn,theexposurecoefficientCeandthethermalcoefficientCt(Note:forCt<1;Ce=1)

mins1,n,t minimumroofsnowloadinkN/m2forloadcase1:uniformdistribution

vm meanwindspeedinm/s

vm,10 meanwindspeedinm/s,measuredin10mheight

w widthofthesnowinthetroughofamulti-pitchedroofafterslidinginm

x horizontaldistancefromtheridgeofanarchedroof(crown:x=0andb=0°)

x30° horizontaldistancefromtheridgetothepointwiththeroofslopeb=30°

Greekupper-caseletters:

Dqe·t accumulatedtemperaturesabovezerodegreesovertimein°C·h

Greeklower-caseletters:

a angleofroofpitch,measuredfromthehorizontal

as roofangleattheslideloadlimit

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as,mean meanvalueoftheslipangleinastandardizedslidingtest

b roofangleofanarchedroof,measuredfromthehorizontal

e emissivityofasurface

qi internalairtemperaturein°C

qe externalairtemperaturein°C

qe,m coldestmonthlymeantemperature

r equivalentdensityofasnowlayerinkg/m3

µb basicshapecoefficientISO-format

µd driftloadcoefficientISO-format

µs slideloadcoefficientISO-format;slideloadlimitvalue

µs,mean meanvalueofthefrictioncoefficientinastandardizedslidingtest

µi shapecoefficientforthelocationiEN-format

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1. Basic principles of EN 1991-1-3 EN 13031-1 and EN 13031-2 follow the basic principles for roof snow loads according to the Eurocode 1, Part 1-3 (EN 1991-1-3), specifying only non-contradictory, complementary information for greenhouses with special properties, which distinguish them from other buildings. However, EN 13031-1 was reviewed and finally published, before the review of EN 1991-1-3 will be finished. Therefore, it is also shown, how EN 13031 can be used with both generations of the Eurocode. For the roof snow loads, EN 1991-1-3:2003 formulates as a basic principle (P): EN 1991-1-3:2003, clause 5.1(1) P and (2): “The design shall recognise that snow can be deposited on a roof in many different patterns.” “Properties of a roof or other factors causing different patterns are:

a) the shape of the roof b) its thermal properties c) the roughness of its surface d) the amount of heat generated under the roof e) the proximity of nearby buildings f) the surrounding terrain g) the local meteorological climate, in particular its windiness, temperature variations, and

the likelihood of precipitation (either as rain or snow).” This is a large and ambitious list. What does the EN deliver? For the calculation of roof snow loads, EN 1991-1-3:2003 is based on a multiplicative format, as opposed to other standards, such as the ISO 4355, which have an additive format, because drifting and sliding snow are added to the basic snow, uniformly distributed. EN 1991-1-3: 2003: Roof snow load: si,n,t = µi · Ce · Ct · sk,n where µi is the shape coefficient for the location i, with µ1 = 0,8 as a reference value for flat roofs with a roof angle a = 0°

Ce is the exposure coefficient Ct is the thermal coefficient sk,n is the characteristic ground snow load* for a reference period n Note: * Exceptional snow loads sAd = Cesl · sk,n can be treated in the same way. The draft versions of the future Eurocode, e.g. prEN 1991-1-3:2020, indicate a shift of the exposure coefficient from the equation for the roof snow load into the shape coefficient µi, where it can be adapted to the shape of the roof. Within the shape coefficient the exposure coefficient Ce can accommodate drift losses as well as drift surcharges better than before. Therefore, this format is also used in EN 13031-1:2019. prEN 1991-1-3: 2020: Roof snow load: si,n,t = µi · Ct · sk,n where µi is the shape coefficient for the location i, with µ1 = 0,8 · Ce as a reference value for flat roofs with a roof angle a = 0° Also, the contents of clause 5.1(1) and (2) remain but are moved to clause 7.1 (1) including a NOTE. Therefore, in 2020 the question remains: What does the EN deliver?

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At least three of the influences of the above-mentioned properties a) to g) have not been included in these formulae or in the three coefficients Ce, Ct and µi provided for the roof snow load:

a) Shape coefficients are provided for some, but only for a few standard roof types: µi b) Heat transmittance of the roof (influence on melting: Ct = f(Uo)): not covered c) Surface material coefficient (influence on sliding) Cm: not covered d) Temperature and heat flux (influence on melting: Ct = f(qi)): not covered e) Drift surcharges are covered for a few, not all roofs types by shape coefficients µi f) Exposure coefficient Ce: National choice, otherwise Ce = 1: not sufficiently defined g) Influence on exposure coefficient Ce: not covered

Rain on snow surcharge or ground snow loads including rain: National choice The missing properties b), c) and d) and the very limited variation of a), e), f) and g) show, that EN 1991-1-3:2003 and also do not provide enough information for heat permeable roof claddings with large heat fluxes due to heating inside (melting of snow) and for smooth or slippery roof surfaces (fast sliding of roof snow). Both physical phenomena, the melting and the immediate sliding of snow, are characteristic for most greenhouse roofs. Many of the specifications provided in EN 1991-1-3, especially those for exposure, drift and sliding have been questioned in the past, see e.g. Sandvik (1996), Schwind (2009), Meløysund (2010). The second edition of the Background reports Part I and Part II refer to the final document EN 13031-1: 2019. Part I about the melting of snow and the use of the thermal coefficient Ct has had few editorial corrections, but no changed content in comparison to the previous edition to the draft prEN 13031-1 from 2017-06-22. However, this second edition of Part II about the sliding and drifting of roof snow contains some major revisions in comparison to the previous edition from 2017-12-27. Text and tables are updated according to the final documents EN 13031-1:2019 and the final draft prEN 1991-1-3:2020. Conclusions for regulations in the Drafts for the National Annexes to EN 13031-1:2019 are drawn. The application of the general rules to specific snow load distributions of different roof shapes are now treated in separate parts, Part II-1 for duo-pitched roofs, Part II-2 for multi-pitched roofs and Part II-3 for arched roofs.

Revisions in Part II-1: For the snow load distribution on unheated single-span duo-pitched roofs the format of the drift surcharge coefficient fd(Ce; a) is adapted to match the newly reviewed Eurocode with suggestions for a national choice of the climate related drift factor d in the National Annex (NA) for EN 13031-1:2019. Revisions in Part II-3: For the snow load distribution on unheated single-span arched roofs a further limitation for the ratio of roof height to snow load h/sk,n is introduced, and the background is updated accordingly. The limitation can also be included in the National Annex (NA) for EN 13031-1:2019.

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Part II-2: Snow load distributions Multi-pitched roofs

9. Multi-span pitched greenhouse roofs in EN 13031-1:2019 9.1 General requirements for greenhouse regulations For the recent review of the second generation of the Eurocode EN 1991-1-3 according to the final draft prEN 1991-1-3:2020, it has been claimed that international codes such as ISO 4355:2013 and ASCE 7-15 have been compared. However, this seems not the case for the regulations for multi-span roofs, as explained in detail in chapter 9.3.1, 9.3.2 and 9.3.3. Also, the background needs to be understood, before any coefficients or values, such as a thermal coefficient Ct or shape coefficients are taken into another standard. Especially the American codes differentiate strongly and are calibrated quite differently. The research for greenhouse related problems in the same documents and others lead to quite different solutions, such as the differentiation of roof snow loads and their distributions according to the thermal conditions (Thermal coefficient Ct) and the surface roughness (Surface material coefficient Cm), see chapter 9.1.2 and 9.1.3. For multi-pitched roofs the redistribution of snow within the limited space of a trough and above it and the related drift losses from the trough are to be specified, see chapter 9.4 and 9.5. The background to general influences on roof snow loads is described in detail in the main background documents Snow EN 13031 Part I: Melting and Part II-0: Sliding and Drift. In this section Part II-2 the main findings are summarized and applied to multi-pitched roofs. 9.1.1 Thermal coefficient Ct EN 1991-1-3:2003 introduces the thermal coefficient Ct but does not specify any values or describe any influences. However, because of the challenges to specify the properties of a large range of different roof claddings and types of roof structures, this is no task for the National Annex of single countries. With the exception of some Scandinavian National Annexes, in the other European countries no further information is given. In a few countries even Ct = 1 is fixed as the only option. This means that greenhouses cannot be designed with EN 1991-1-3 + NAD alone. Here the responsibility of the greenhouse Standardisation begins. The direction for further investigations is also shown, using the ISO 4355 based on international research beyond Europe, e.g. from Japan, Canada and the USA. The draft versions of the future Eurocode, e.g. prEN 1991-1-3:2020, show no sign of basic changes concerning snow melting or sliding. However, the reference to ISO 4355 in the existing norm was replaced by a paragraph to the choice of the thermal coefficient Ct. prEN 1991-1-3:2020, 7.4 (2) NOTE 1: “Locations where the duration of the snow load is long enough can be selected on the basis of the characteristic ground snow load greater than a threshold value sk,min. sk,min = 1,5 kN/m2 unless the National Annex gives a different value for use in a country.” A characteristic snow load value of 1,5 kN/m2 is no limit for the application of a thermal coefficient Ct. It is just the smallest characteristic snow load in Norway.

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Paragraph 7.4 is no sufficient description of the conditions, application rules and safety measures required for the application of Ct < 1 for greenhouse roofs. These rules in connection with a “controlled heating operation” are specified in EN 13031-1:2019 and will also be found in the second part EN 13031-2 with the background given in Part I of this Background report. There are two types of conditions for the application of Ct < 1 on greenhouse roofs. The first group is for physical limits and safety related requirements, the second group concerns the correct calculation of a thermal coefficient with its limitations depending on the background (Sandvik, 1988). ISO 4355, Annex D and recently prEN 1991-1-3:2020 misinterpret some of this background, for example the limit of 1,5 kN/m2, the upper limit of Uo or the roof angle influence. To be able to use the thermal coefficient Ct according to EN 13031-1:2019, Annex C for the reduction of the roof snow load on greenhouse roofs there are five requirements:

1. Transparency of the roof cladding In contrast to other buildings, on greenhouse roofs snow needs to be removed as fast as possible, to allow the plants to survive. That is why the first and central requirement for Ct < 1 is the transparency of roof (and wall) cladding. Transparency itself does not contribute to the physical processes of snow melting. However, a side effect of a transparent cladding is, that snow and ice remain visible, to allow the removal of any excess snow, or if this is not possible, to evacuate the area. The risk can be located and assessed visually under a transparent roof. This is not possible for non-transparent roofs.

2. Heat permeability of the cladding (Thermal transmittance Uo ≥ 1 W/(m2K) The lower limit of Uo = 1 W/(m2K) is generally accepted as sufficient for normal buildings, see ISO 4355, Annex D, equation (D.2), EN 1991-1-3:2003, clause 5.2(8) and prEN 1991-1-3:2020, clause 7.4. In case of doubt, especially if the cladding is advertised as heat protective, the heat transmission requirement of Uo ≥ 1 W/(m2K) must be checked by a calculation or test, see ISO 4355, EN 673, EN 674, EN 675, EN ISO 6946, EN ISO 10077-1 and EN ISO 10077-2. That can be required for multi-layer glass or plastic with reflective layers, additives, surface structure or coating. However, the limit is no physical limit. Snow also melts on roof surfaces with Uo < 1 W/(m2K). Thermodynamic melt rates depend on the product Uo · DT with the gradient DT between internal air temperature qi and the melt temperature of snow around qsm = 0°C. 3. Sufficient heating min qi >> 5°C (internal air temperature under the roof) Because the melting of snow depends as much on the temperature gradient as on Uo, this limit is as important. The greenhouse heating must be able to maintain an internal air temperature qi under the roof during and after snowfall, with a defined minimum internal air temperature min qi well above 5°C and preferably up to 18°C, until the snow is removed. This is called a controlled heating operation. During this time, ventilators should be closed and the screens, also the thermal screens, should be opened or retracted. When a gutter heating (snow melting tubes near the gutter) is used, the screens under the gutter heating can remain closed. That increases the air temperature directly under the roof. The specification of min qi = 12°C or min qi = 18°C, is a national choice depending on the snow climate, operational experience and technical equipment of the users. For characteristic snow loads sk,50 above 1,5 kN/m2 or more than 50 mm SWE (Snow-Water-Equivalent) of new

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snow per day, a heating operation with min qi = 18°C is recommended, otherwise min qi = 12°C can be used. This is proven safe for commercial production greenhouses. 4. Unrestricted melt water outflow The fourth requirement for a roof snow load reduction is, that the melt water can be drained quickly without risk of icing, of bridging internal gutters or clogging internal drains. Here the experience of the greenhouse producers and the gutter design play a role. In cold winter climates, a gutter heating can be supportive. If a gutter is designed sufficiently for maximum precipitation rates (mostly in the summer half year) and clogging is prevented, by experience it is also sufficient for the melt water outflow in winter.

5. Additional safety measures Further safety measures can be chosen nationally. Greenhouses may only be considered as heated (use Ct < 1), if an automatic back-up system is present, capable of taking over the heating, before the temperature drops below 5°C. That should include an emergency electrical power supply. Qualified climate and weather surveillance with alarm system should be available, e.g. provided by so-called climate computers.

The Ct-formula according to ISO 4355, Annex D is based on a thermodynamic model for the accumulation and ablation of roof snow with melting at the bottom by the meteorologist Rune Sandvik (1988), using statistical data of snowfall events in Norway. Not snow heights, but precipitation heights, available for 12-hour measurement intervals together with other data, corrected for drift losses by wind, evaporation losses and surface wetting, have been used. This statistical approach has the advantage that other methods may not have, e.g. CSA S367-9 or Rees et.al. (2002) and Liu et.al. (2006a, 2006b) which are all based on snowfall rates. However, design values for snowfall rates are hardly available everywhere. For heated greenhouses equipped for the controlled melting of roof snow, the thermal coefficient Ct in accordance with ISO 4355, Annex D can be used, completed by the missing roof angle function f(a). On the safe side for f(a) only the roof area is considered. Ct = { 1 - f (sk,n) · f (qi) · f (Uo) } · f (a) Ct = {1 - 0,054 · (sk,n / 3,5)1/4 · (qi - 5°C) · [sin((0,4 · Uo - 0,1)·180°/p)]3/4} · cos(a) With the following influences:

• Snow load (correlated to snowfall rate): f (sk,n) = 0,054 · (sk,n / 3,5)1/4 • Inside air temperature (heating): f (qi) = qi - 5°C • Heat transmittance roof: f (Uo) = [sin ((0,4 · Uo - 0,1)·180°/p)]3/4 • Ratio of ground area to roof area: f (a) = cos (a)

The mathematical and statistical formulation of this equation requires the following limitations:

a) Air temperature inside: min qi = 5°C ≤ qi ≤ 18°C = max qi b) Heat transmittance: min Uo = 1 W/(m2K) ≤ Uo ≤ 4,177 W/(m2K) = max Uo c) Air temperature outside: If qe < -8°C, take Ct = 1,2 · Ct ≤ 1 with qe - mean temperature

during snowfall period d) Snow load (upper limit): If Ct < 0,2 · f (a), take Ct = 0,2 · f (a) with f (a) - roof angle

function. Upper ground snow load limit: lim sk,n = = { 20,2634 / [ f(qi) · f(Uo) ] }4 e) Snow load (lower limit): For Ct < 1 a lower limit for the roof snow load after melting of

min s1,n,tshould be applied. The limit can be chosen nationally (NDP), otherwise the limit is min s1,n,t=0,25 kN/m2.

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The background of these limitations and further details about the melting of snow on greenhouse roofs is given in the Background report Part I: Melting. The limits can be explained as follows: a) The lowest value of 5°C internal temperature is a reserve, to ensure, that the snow melting can

start. Above 18°C, there is no experience (melt rate, documentations, data). Only a few plants tolerate very high temperatures, e.g. Orchids. The limit of 18°C is used for design purposes and it is a robustness measure.

b) The sine-function for the simulation of the influence of the heat transfer coefficient f(Uo) has

mathematical limits. The influence function of the heat transfer coefficient f(Uo) reaches its minimum for the minimum value of the sine-function: sin (0,4 · Uo - 0,1) = 0, giving the lower limit value of min Uo = 0,1 / 0,4 = 0,25 W/(m2K). The sine-function has also a maximum, giving the exact upper limit of max Uo = 4,177 W/(m2K). This limit increases the robustness of the formulation. Single layer claddings used for commercial production greenhouses have much larger Uo. The upper limit of 4,5 W/(m2K) given in ISO 4355, Annex D, is not correct.

c) There are also conditions for the climate when using the Ct- formula. These limits seem to have

physical reasons. It should not be too cold, otherwise the energy losses increase. In the evaluations by Sandvik min qe = -8°C was identified as a limit, where Ct needs to be enlarged.

d) The snow load function f(sk,n) = 0,054 (sk,n/3.5)1/4 correlates the climate (and its ground snow

load sk,n) to the snowfall rate or snowfall intensity and duration. Higher ground snow loads are the result of colder temperatures with longer accumulation periods and very often more than one snowfall event, not necessarily of much larger snowfall rates. That is why, for larger sk,n the Ct-values tend to get smaller. However, for very large sk,n the value of Ct is reduced faster than the ground snow load increases, leading to decreasing roof snow loads. To avoid this effect, the upper limitation for the ground snow load lim sk,n = { 20,2634 / [ f(qi) · f(Uo) ] }4 is introduced. It refers to a lower limit for the thermal coefficient at min Ct = 0,2 · f (a). Smaller Ct or larger sk,n should not be used with this formula.

e) The lower snow load limit min sk,n,t = 0,25 kN/m2 is a robustness measure, based on practical

experience in many European countries. It is a national choice (NDP) and should be used instead of a limit of 1,5 kN/m2 for the smallest ground snow load in Norway.

9.1.2 Surface material coefficient Cm In a 20-yearlong measurement campaign (1965-1986) on 200 freestanding unheated gable roofs of farm buildings by Høibø (1988) in Norway with over 1300 measurement data, a clear difference in roof snow loads has been recorded between metal roofs (Cm = 1,2) and other claddings such as brick or concrete tiles, asphalt roofing or wood shingles (Cm = 1). The surface material coefficient Cm in ISO 4355 is based on that. However, warm roofs and roof surfaces more slippery than the trapezoid plates of metal on farm buildings were not covered in the measurements by Høibø (1988). Standardised sliding tests according to Jelle (2012), can help to classify the surface roughness of modern roof claddings. Table 1 shows preliminary slip angles as,mean, when the snow and ice probes start to slip downwards, and the appropriate friction coefficients µs,mean = tan as,mean for different surface types. The results for brick tiles and for polymeric roofing have been questioned. There is evidence that ETFE- or PC-plastic film used for greenhouse roofs is usually more slippery than glass (Ito et.al. (1997)). The slightly convex shape of the plastic film when filled with air contributes to an even faster sliding of snow (Terasaki & Fukuhara (2016)).

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Type of surface Slip angle

as,mean in ° Friction coefficient

µs,mean Surface Roughness

Classification Glass 0,1 0,002

Slippery Cm = 1,333

Roof window 0,12 0,002 Solar cell panels 0,17 0,003 BIPVs (Building Integrated Photovoltaic) 0,19 0,003 Solar thermal panels 0,94 0,02

Smooth Cm = 1,2 Steel or metal plates 2,7 0,05

Brick tiles (with smooth surface?) 3,5 0,06 Polymeric roofing (check?) 16 0,3

Rough Cm = 1

Slate tiles 23 0,4 Granulated steel 28 0,5 Rough concrete tiles 31 0,6 Granulated bitumen roofing 36 0,7 Note: The given values are examples, for other probes (density, size) other values may appear. The tests are carried out under ideal conditions with rectangular packed snow probes. That is why the friction coefficients are not to be applied to natural roof snow covers. They are only shown here as a numerical scale for the surface roughness.

Table 1: Slip angles and snow friction coefficients according to Jelle (2012) For different types of greenhouse cladding, as shown in Table 1 and classified in Table 2, the surface material coefficients Cm could be uses in accordance with ISO 4355, clause 6.3. The conditions for the choice of Cm are related to the thermal coefficient Ct and the physical conditions for melting or adfreezing of snow. EN 13031-1:2020, Annex C, C.1(2) introduced the following application for greenhouses: • Cm = 1,333: for warm slippery unobstructed surfaces with controlled heating according to C.2.1

(transparent; internal temperature min qi >> 5°C, Ct < 1; heat transmission Uo ≥ 1 W/(m2K)). • Cm = 1,2+: for other warm smooth, unobstructed surfaces according to Table C.1 (transparent;

min qi ≥ 5°C, Ct = 1; as a rule, above a closed space). • Cm = 1,2: for cold smooth unobstructed surfaces according to Table C.1 (non-transparent or

transparent, cold smooth unobstructed surfaces (min qi < 5°C)). • Cm = 1: for other cases (cold rough surfaces, open, exposed or ventilated roofs (min qi < 5°C). Note +: For Cm = 1,2+ drift surcharges can be neglected as for Cm = 1,333. The roof has to be warm despite of Ct = 1. However, if drift surcharges are possible, Cm = 1,2 should be used for cold smooth unobstructed surfaces, as cold metal roofs were part of the roof snow load investigation by Høibø (1988).

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9.1.3 Classification of greenhouse roofs in EN 13031-1:2019 To support the choice of thermal coefficient Ct und surface material coefficient Cm and to identify the suitable snow load distribution for the greenhouse, EN 13031-1:2019 introduced the classification shown in Table 2. This is a recommendation and guidance only. The values in grey fields (boxes) are open to further national regulations (NDP).

Type of greenhouse / greenhouse compartment / roof / climate and exposure

Slippery roof cladding / closed building Others T: Transparent cladding a, e.g. glass, plastic film or membrane, plastic sheets with smooth surface, BIPV f

NT: Non-transparent e

not heated e.g.

metal plates, sandwich,

textile membrane,

BIPV e

Non-slippery surface: e.g. tiles,

granulated surfaces Open,

exposed or ventilated

roof: e.g. porch,

canopy, mounted PV

H: Heated with minimum air temperature under the roof:

min qi in °C

NH: Not

heated S: Heat transmittance sufficient:

Uo ≥ 1 W/(m2K) NS: Heat

transmittance not sufficient: Uo < 1W/(m2K)

e.g. heat protective cladding

C: controlled heating b

F: frost-free c

min qi >> 5°C min qi ≥ 5°C min qi < 5°C

Thermal coefficient Ct and Surface material coefficient Cm according to ISO 4355

Cm = 1,333 Cm = 1,2+ d Cm = 1,2 Cm = 1

Sliding without drift surcharges - Sliding with drift surcharges

Ct < 1 Ct = 1 a T (transparent cladding): solar radiation transmission possible, required for the growth of plants. Side effect: Snow accumulation remains visible, so that immediate measures can be taken to remove snow and restore full radiation transmission. b C (controlled heating): Heating device is intended to melt a required amount of snow, to keep the roof snow load limited and remove all snow from the roof as fast as possible. c F (frost-free): Heating device is not intended to melt snow, but keeps the temperature just above the required limit for optimum crop production or survival, at least qi = 5°C. d NH (not heated): No heating available. In warm and moderate climate with snowfall temperatures around 0°C where sliding of snow is more likely than snow drifting: Cm = 1,2+. In cold climates where drift surcharges are possible, the surface material coefficient Cm = 1,2 should be used. e NT (non-transparent cladding, not heated): The area with non-transparent cladding is an integrated and minor part of the greenhouse area, meant for specific functions, directly related to the professional production of plants and crops, such as heating, storage, water management, etc. f BIPV (Building Integrated Photovoltaic) roof cladding, in contrast to roof-mounted PV Table 2: Greenhouse categories due to cladding, heating, climate and exposure according EN

13031-1:2019; Annex C, Table C.1 Note, that for non-transparent claddings (NT) as for others the heating cannot be taken into account.

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9.2 Measurements on multi-span roofs The most severe drifting on a multi-pitch roof was measured in a Survey of the National Research Council of Canada on an unheated airport building in Ottawa, reported by Taylor (1980) and many other authors after him. From the 3rd to the 5th of March 1971, the Eastern Canadian Blizzard was one of the four most severe “storms of the century”. The blizzard produced up to 73 cm of new snow on March the 2nd and continued with 40,6 cm at wind speeds up to 61,4 m/s on March the 4th. After 3 days, a ground snow load of 3,35 kN/m2 was attained in Ottawa. At the end of the season 444 cm of snow had piled up, “the most in the city’s record”. The multi-span roof had 3 spans of 11 m each and a roof height of 1,83 m between eaves and ridges. The eaves height was small with 3 m above ground. The roof angle of this building was also very low with 18,4°, giving a ridges height of 4,83 m. With these dimensions the building remained within the ground snow drift zone of up to 5 m according to Tabler (2003), see also Part II of this Background Report. The rather exposed airport location may also have provided fetch distances of 150 m to 300 m to develop a saturated drift flux (Takeuchi (1980)). Therefore, the drift conditions were unusual for roofs of normal buildings but otherwise ideal. Because of the severe snowstorm the snow load distribution due to drifting snow has nearly reached the equilibrium state, where the drifted snow surface is streamlined enough for the wind flow to get reattached to the surface. After that, drift growth on the roof stops and the snow load can only be enlarged by further non-drifting snowfall. On this multi-span roof, the snow would continue to accumulate in the same way as on a flat roof. The near-equilibrium distribution in Figure 1 is shown together with the ground snow load level, as it would appear on the roof without drift losses by wind. On the roof, a mean snow density of 300 kg/m3 was measured after the blizzard. This is a smaller drift snow density than predicted by Tabler (2003), because it was the result of a single snowfall event of no more than 3 days. With the mean snow density of 300 kg/m3 (2,943 kN/m3), the measured ground snow load of 3,35 kN/m2 would have resulted in a snow height of 1,117 m above the roof surface, if there were no drift losses. This amount of snow was about 22% larger than the volume of the trough. Therefore, some snow above the ridges was to be expected.

Figure 1: Incomplete drifting on a multi-span roof Ottawa 1971 according to Taylor (1980)

The measured snow load distribution in Figure 1 shows that not much snow settled in regions with high wind speeds, the first windward roof pitch and the upper half of the windward roof pitches of the second and third roof span. Most snow has accumulated behind the first ridge, where the air flow was separated from the roof surface and had to slow down. A drift formed in the zone with reverse air flow on the leeward side. From there the drift did grow to the windward side of the valley filling it nearly up to the ridge. The air flow may have reattached itself to the new surface, only slight discontinuities below the second and third ridge are left.

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The same drift growth processes happened in the second valley, but with a smaller wind speed and a smaller drift rate. However, the wind was still strong enough to create a nearly ideal horizontal roof snow surface on the third and outer leeward roof face. It has to be noted, that the roof snow did not end vertically above the outer edges, it was overhanging and started to creep down. Also, the large wind- and leeward drifts at the ground are not shown in Figure 1. It is important to understand, that the snow was not “redistributed” by wind after it accumulated, as often claimed, the drifting snow settled there at the first place due to the prevailing wind field saturated with drifting snow. More snow settled down in areas with lower wind speed, changing the wind field in such a way, that the wind could flow with less resistance over the obstacle. Drifting is a self-regulatory process. In total the roof snow after drifting was not larger than the ground snow level, as it is shown above the roof surface in Figure 1. This is also shown by calculation. The measured and calculated data and ratios are summarized in Table 3.

Span 1 Span 3 Span 3 Windward side Valley 1 Valley 2 Leeward side Ridges: 1 2 3

windward leeward windward leeward windward leeward Measured roof snow heights in cm: 61 & 41 36 122 & 155 185 114 & 41 23 97 160 94 20 89 & 155

Measured mean density roof snow: 300 kg/m3 Measured ground snow load at the site: 3,35 kN/m2 Ground snow height: 112 cm

Global snow load values calculated from values above: Roof snow load in kN/m (per m roof length):

7,634 22,034 11,86 16,078 10,313 15,054 Total roof snow load: 82,97 kN/m roof length

Ground snow load in kN/m (per m roof length): 18,425 18,425 18,425 18,425 18,425 18,425

Total ground snow load: 110,55 kN/m roof length Ratio of roof snow load to ground snow load:

0,414 1,196 0,644 0,873 0,5597 0,817 0,414 0,9198 0,716 0,817

0,805 0,758 0,688 Total ratio of roof snow load to ground snow load: 0,75

Local values at ridges and valleys: Roof snow load in kN/m2:

1,056 5,428 0,675 4,964 0,587 Ratio of roof snow load to ground snow load sG = 3,35 kN/m2:

0,315 1,62 0,201 1,401 0,175 Table 3: Measured roof snow on a cold multi-span roof with a low roof angle (18,4°) in

Ottawa after the “Blizzard of the century” March 1971 Because of the low roof angle and the fact, that more snow arrived than the valleys would accommodate, there was still 36 cm, 23 cm and 20 cm snow above the ridges. On the windward side of the first span the snow height is small, but the roof not cleared. For higher roof angels than 18,5°

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and a larger roof valley or slightly less snow, the windward side of the first span and the ridges would have been cleared of snow. From CFD investigations it is known that higher wind speeds above 4 m/s and roof angles above 17° to 20° are main influences (Tominaga et.al. (2016)). The roof surface roughness and roof temperature might also be highly influential. These limits are still controversial. Therefore, it is a suitable approach for prEN 1991-1-3:2020, to assume complete windward sliding for all pitched roofs a > 5°. Other codes differentiate and might have other limits such as 15° according to NBCC. However, such limits should be consistent within a standard, which is not the case in prEN 1991-1-3:2020. With the given snow heights measured at 2 to 4 points along the roof pitches a total roof snow load of 82,97 kN per meter roof length can be calculated. The measured ground snow load of 3,35 kN/m2 after 3 days only resulted in a snow height of 1,85 m in the first valley being slightly larger than the roof height of 1,83 m. In the second valley the snow height 160 cm is 86,5% of the maximum height. The total roof snow load is only 75 % of the total ground snow load, not more, as often implied in publications. Only the first leeward roof pitch had 20% more snow than the ground snow value of the roof pitch might have. This was an intermediate drift state and the snow would have been distributed more evenly in the equilibrium state with a little more wind. The first span had in total 80% and the first valley 92% of the ground snow level. The local snow load value in the first valley (1,85 m roof snow) was not larger than 1,62 times the ground snow value. For a comparison, a flat roof in the same exposed airport location in Canada would have to be designed for a roof-ground snow load ratio of 60% using a basic roof snow coefficient of Cb = 0,8 and a drift coefficient of Cw = 0,75 according to NBCC. Although this short-time snowfall event did not represent the characteristic snow load level, it showed that on multi-span roofs a little less snow is removed by wind than on flat roofs. Conclusion: It can be seen that in the equilibrium state for the unbalanced load case with unlimited wind and snow supply for multi-span roofs the snow surface would be parallel to the ridge heights with a snow-free windward roof pitch, a triangular distribution on the outer leeward roof pitch and zero snow on all the ridges (s0 = 0). The maximum value above the gutter would be s3,max = g · h. In this specific limit state only a certain amount of snow can be contained within the limited space of the valley. Assuming there are no drift losses but losses by melting snow, the snow load potential is sk,lim = (g · h/2) / Ct. Potential drift losses are a national choice. If drift losses from a valley are to be allowed, they should be smaller than on a flat roof as a reference and limited by the overall roof size as well as the number of valleys in a row. The equilibrium state is a limit for usual design snow load distributions for non-equilibrium stages when there is not enough snow (sk < sk,lim) or too much snow (sk > sk,lim) available due to snowfall. Different than in prEN 1991-1-3:2020, the limit cannot be described by a maximum value for the shape coefficient µ3 alone, the total snow load in the valley is limited. Also, it does not depend on the actual snow load sk, which can be smaller or larger. For the format according to prEN 1991-1-3:2020 with si = µi · Ct · sk the snow load potential and shape coefficients for the limit can be given as follows: Equilibrium limit state for the snow load distribution on multi-pitched roofs:

Snow load potential to fill the valley completely: sk,lim = (g · h/2) / Ct Above the ridge (i = 0): s0,min,lim = 0 µ0,lim = 0 Above the gutter (i = 3): s3,max,lim = g · h µ3,lim = 2

Where Ct is the thermal coefficient; h is the depth of the valley in m; g is equivalent weight density of the accumulation in the valley * in kN/m3

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Note: * The equivalent density is a national choice. In prEN 1991-1-3:2020 g = 2 kN/m3 is recommended for cold roofs with Ct = 1. For warm roofs with controlled heating according to EN 13031-1:2019 g = 3,5 kN/m3 should be applied.

For snow loads below the limit (sk < sk,lim) any distribution can be chosen as long as it gives a smooth transition to the equilibrium state as in nature, see measurements in Figure 1. For warm roofs with slippery roof surfaces complete sliding should be considered with µ0 = 0. For cold roofs and for small roofs, where sliding is restricted, incomplete stages of redistribution can be chosen. However, for snow loads above the limit (sk > sk,lim) the excess snow above the ridge level is distributed as on a flat roof. The drift losses should be consistent with the flat roof model. Grammou (2015, 2019) investigated snowdrifts on photovoltaic (PV) panels mounted on flat roofs. The water channel that was used, with fluid characteristics and sand particles matched natural drift conditions better than artificial snow in a wind channel with small models. The water flume model was calibrated in such a way as to produce similar distributions as snow in the moderate climate of the middle of Europe (80% of German territory). Small fluid speeds were sufficient to “blow” the ridges of the PV panels free of sand. The largest measured shape coefficient in the valleys between the PV panels was µ2 = 1,57 < 1,6 for a distribution with plateau and µ2 = 2 for a triangular distribution. The accumulations in the middle were larger than at the edges, where the sand could be removed completely with increasing duration (equilibrium). An important finding not only applicable to roof mounted solar panels was, that in all of the tests there was less sand on the roof model than on the bottom of the water channel. It was shown, that snow is preferably blown down and does not settle in drift mounds above ridge level. The height of the solar panels forms a hard barrier as solid snow fences would do (Tabler (2003)). Extreme unbalanced snow accumulations on a roof are produced together with large drift losses. Snow drifts are shaped like the wind profiles that formed it.

9.3 Snow load distributions - Models in different standards 9.3.1 Multi-span roofs in the Eurocode – present and future International research shows, that drifting cannot produce more snow on the roof than on the ground, as the drift models in EN 1991-1-3:2003, chapter 5.3.4 or in prEN 1991-1-3:2020, chapter 7.5.4 imply, with up to 144% and now even up to 154% of the ground snow load on the roof (for a = 30° and Ce = 1,2; Ct = 1,2). The recent model in the Eurocode EN 1991-1-3:2003 seemed to aim just on using the roof angle influence as for other pitched roofs. In the trough the snow loads increased locally up to the magic limit of µ = 1,6 = 2 · 0,8 in the unbalanced load case (ii). That was all. Between 5° and 30° despite the increasing snow load in trough, the snow load at the ridges remains unchanged. For 30° the total roof snow load above the trough is 144% of the ground snow load (Ce = 1,2). For the balanced load case (i) the snow used to disappear above the trough with roof angles large enough, see Table 4. The new regulation in prEN 1991-3:2019 shown in Table 4 is also no redistribution model. For the ridges the same roof angle function is used as for other pitched roofs. The complete failure for larger snow loads in comparison to the trough height and large roof angles is only just avoided by three measures. 1) With µ2,b(30°; 1) = 0,8 in load case (i) the snow load remains at 80% of the ground snow level despite of the roof angle. 2) The maximum shape coefficient µ3(a) in the trough is now limited when reaching the trough height h by µ3(a) ≤ g · h / sk. 3) The maximum µ3,max can be chosen nationally (NDP), otherwise µ3,max = 1,6 applies as before.

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However, despite of the three measures probably taken from the new German National Annex DIN EN 1991-1-3/NA:2019 and partly because of it, the model remains inconsistent. Without load case (i) it would fail completely and become unsafe for large ratios of snow load to roof height and for large roof angles. The reason is simply, that the snow is not redistributed within the limited space of the triangular trough and if the trough is full above the ridges, despite the fact, that this is very easy, just trigonometry.

EN 1991-1-3:2003 Shape coefficients

Shape coefficients µ2 (a) and µ3 (a)

0° ≤ a ≤ 30° 30° < a < 60° a ³ 60°

Load case (i) µ2 = 0,8 µ2 = 0,8 (60° - a) / 30° µ2 = 0

Load case (ii) µ3 = 0,8 + 0,8 a / 30° µ3 = 1,6 -

Format Roof snow load: s = sk · Ce · Ct · µi Where a is the roof angle (inner trough area µ3 is calculated with a = 0,5 · (a1 + a2)); µ3 is the maximum shape coefficient above the gutter; µ2 is the shape coefficient above the ridge depending on the roof angle.

prEN 1991-1-3:2020 Shape coefficients

Load case (i)

Balanced Load Arrangement

Uniform distribution for roof angle a in °

Windward Above the troughs Leeward

µ2,b (a1, Ce) µ2,b (a = 30°; Ce = 1) = 0,8 µ2,b (a2, Ce)

Basic shape coefficient for outer roof surfaces ai for i = 1 und 2:

as

Saddle roof

5° < ai ≤ 30° µ2,b (ai; Ce) = 0,8 Ce

30° < ai < 70° µ2,b (ai, Ce) = 0,8 Ce · (70° - ai)/40°

ai ³ 70° µ2,b (ai, Ce) = 0

Load case (ii)

Unbalanced Load Arrangement (outer roof surfaces as case (i))

Above the ridges: Shape coefficients as for outer roof surfaces Above the gutter: µ3 (a) = 0,9 + 0,7 · a/30° ≤ g · h / sk ≤ µ3,max

Format Roof snow load: s = sk · Ct · µi Where a1,a2 is the roof angle of the roof pitches; inner troughed area: a = 0,5 · (a1 + a2); h is the height of the trough in m; g is the density of the snow with g = 2 kN/m3, if not otherwise specified (NDP) µ3,max is the maximum above the gutter with µ3,max = 1,6, if not otherwise specified (NDP)

Table 4: Shape coefficients and snow load distributions for multi-span pitched roofs according EN 1991-1-3:2003+A1:2015 and prEN 1991-1-3:2020

The model is also not calibrated for all cases of multi-span roofs it is supposed to cover. No comparison to flat roofs, small or large, is made. International research and the appropriate regulations in other modern snow load standards such as ISO 4355:2013 or ASCE 7-10 are ignored. How problematic this regulation can get, especially for large roof areas with many small roof spans in comparison to the snow load, can be demonstrated on the shape coefficients and the ratio of the total snow load in the trough to the ground snow load. For a large multi-span roof with equal roof heights and roof angles the total load above the trough can be calculated as follows:

S STrog = sk · Ct · ½ { µ2,b(a; Ce) + MIN (µ3(a); g · h/sk; µ3,max)) }

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Where Ct is the thermal coefficient and Ce is the exposure coefficient; a is the mean value of the roof angles within the trough: a = ½ (a1 + a2); µ2,b(a;Ce) is the basic shape coefficient above the ridge (see Table 4); µ3(a) is the shape coefficient above the gutter: µ3(a) = 0,9 + 0,7 · a/30° ≤ g · h / sk ≤ µ3,max g is the density of the snow g = 2 kN/m3 (National choice by NDP);

h is the height of the roof (depth of the trough) in m; µ3,max is the maximum µ3,max = 1,6 (National choice by NDP)

In Table 5 the results for normal cases with Ct = 1 und Ce = 1 are shown. Because of the wrong roof angle influence in the model due to the ignored principle, that snow has to be redistributed within a limited space, the total snow loads vary between 80% and 120% of the ground snow load level. For higher snow loads in comparison to the roof height and for larger roof angles the total snow load decreases down to the level of a small flat roof, because of load case (i).

Roof pitch angle a in °

Shape coefficients (for Ce = 1)

Representative value including limits for shape coefficient µ3 lim µ3 = g · h / sk ≤ 1,6 with g = 2 kN/m3

Characteristic snow loads on the ground sk in kN/m2 µ2,b µ3 0,65 kN/m2 1 kN/m2 1,5 kN/m2 2 kN/m2 4 kN/m2 6 kN/m2

g · h / sk = 3,077 2 1,333 1 0,5 0,333 10° 0,8 1,133 1,133 1,133 1,133 1 0,5 0,033 20° 0,8 1,367 1,367 1,367 1,333 1 0,5 0,033 30° 0,8 1,6 1,6 1,6 1,333 1 0,5 0,033 40° 0,6 1,833 1,6 1,6 1,333 1 0,5 0,033 50° 0,4 2,067 1,6 1,6 1,333 1 0,5 0,033 60° 0,2 2,3 1,6 1,6 1,333 1 0,5 0,033 70° 0 2,533 1,6 1,6 1,333 1 0,5 0,033

If the trough is full: lim µ3 ≤ 1,6 is used instead of µ3

Case (i) Ratio roof to ground snow load a in ° µ2,b s2/sk = Ct · Ce · µ2,b with: µ2,b(30°; 1) = 0,8 above the trough; for Ct = 1

10° - 30° 0,8 0,8 40° 0,6 0,8 50° 0,4 0,8 60° 0,2 0,8 70° 0 0,8

Case (ii) Ratio total snow load in the trough to ground snow load

a in ° µ2,b sTrog/sk = Ct · (µ2,b + MIN(µ3; lim µ3; 1,6))/2 with Ce = Ct = 1 10° 0,8 0,956 0,956 0,956 0,9 0,65 0,417 20° 0,8 1,083 1,083 1,067 0,9 0,65 0,417 30° 0,8 1,2 1,2 1,067 0,9 0,65 0,417 40° 0,6 1,1 1,1 0,967 0,8 0,55 0,317 50° 0,4 1 1 0,867 0,7 0,45 0,217 60° 0,2 0,9 0,9 0,767 0,6 0,35 0,117 70° 0 0,8 0,8 0,667 0,5 0,25 0,017

Table 5: Shape coefficients, limits for µ3 and ratio of roof to ground snow load for an example for roof height h = 1 m (Ce = 1; Ct = 1) according prEN 1991-1-3:2020

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On the example of a multi-span roof with symmetrical roof troughs with 30° roof angle (µ2,b = 0,8 Ce), the extreme trends can be shown, when the exposure coefficient and the thermal coefficient are varied as in Table 6. For large snow heights the model is less safe than for small snow heights. This is the opposite trend in comparison to other roofs.

Ratio of snow height

to trough height

Typ of building (thermal

Influence)

Thermal coefficient

Ratio of roof to ground snow load Exposure coefficient Ce

1,2 1 0,8

Small: µ3 relevant

Freezer halls, Ice sport halls Ct = 1,2 1,536

(µ2,b = 0,96; µ3 = 1,6) 1,44

(µ2,b = 0,8; µ3 = 1,6) 1,344

(µ2,b = 0,64; µ3 = 1,6)

Normal case Ct = 1 1,28 (µ2,b = 0,96; µ3 = 1,6)

1,2 (µ2,b = 0,8; µ3 = 1,6)

1,12 (µ2,b = 0,64; µ3 = 1,6)

Large: lim µ3 < 1,6

relevant

Freezer halls, Ice sport halls Ct = 1,2 Case (i): µ2,b = 0,8: S S/W/sk = 0,96

(µ2,b = 0,8 · Ce; µ3 = g · h/sk see Table 13: Case (i) relevant)

Normal case Ct = 1 Case (i): µ2,b = 0,8: S S/W/sk = 0,8 (µ2,b = 0,8 · Ce; µ3 = g · h/sk see Table 13: Case (i) relevant)

Table 6: Ratio of roof to ground snow load (S S/W) / sk and shape coefficients for the troughs of multi-span roofs according to prEN 1991-1-3:2020

Conclusion for greenhouses: The model for multi-span roofs in prEN 1991-1-3:2020, Figure 7.6 shows just two joint gable roofs with one valley between them, a duo-span situation. The snow loads seem to aim at large roof height to snow height ratios. The sliding of roof snow and the redistribution of snow in a small limited space is not considered. There are safety issues.

Figure 2: Sliding of snow on a greenhouse Type Venlo with plastic film after an unexpected

snow fall event (during construction: emergency heating up to 7°C) For typical multi-span pitched greenhouse roofs with many small roof troughs with small ratios of roof height to snow height the models of EN 1991-1-3:2003 and prEN 1991-1-3:2020 are not suitable. Therefore, the greenhouse standard EN 13031-1:2020 has specialized regulations for the type of roofs, it covers. They are rather based on other modern snow load standards, such as ISO 4355:2013 and ASCE 7-10, but avoiding their shortcomings, which will be discussed in chapter 9.3.2 and 9.3.3.

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9.3.2 The redistribution model in ISO 4355:2013 The drift model for multi-pitched roofs in ISO 4355:2013, Annex B, chapter B.3 is based on the redistribution of the basic roof snow load, adapted by the general conversion factor of fc = 0,8, the exposure coefficient Ce and the thermal coefficient Ct, but for a basic shape coefficient of µb = 1. In this limit state, the snow is allowed to slide down completely (sliding load case 2 in ISO 4355:2013), but only after age related losses due to drift, melting, sublimation / evaporation have occurred. If the snow quantity is too large for the roof trough, the excess snow is distributed evenly above the ridge level (sliding load case 1 in ISO 4355:2013). For the snow after sliding a mean density of r = 300 kg/m3 is recommended. ISO 4355:2013 formulates two principles:

• “However, for multi-pitched roof, the snow slides and results in a redistribution of load on the same roof.”

• “The sliding load case accounts for the potential sliding snow and possible drifted snow.” ISO 4355:2013 seems to consider a final state distribution. Therefore, drift losses as large as for a flat roof are allowed. This should be questioned. Also, the influence of the roof size is not mentioned. For large multi-span roofs, an increased exposure coefficient Ce ≤ 1,25 as for large flat roofs could be used. With that the ground snow load level would be reached on the roof. However, such a regulation would also be not suitable if the thermal coefficient Ct < 1 according to ISO 4355, Annex D, which is based on precipitation data and is to be used with µb = 0,8 and Ce = 1. For smaller roofs it remains questionable, whether the general conversion factor and exposure coefficient Ce in the formulae can have the same influence as on a flat roof. Drifting can reduce snow on the ridges, but most of the drifted snow will fall back into the troughs and not be removed from there. Also, sliding from the ridges down into the trough would reduce the possibility for further drift losses. As for multi-level roofs in ISO 4355:2013, Annex B, chapter B.5, a roof trough could be treated as a sheltered zone below the ridge: Ce0 = 1,2. For snow on the ridges and above ridge level Ce0 < 1,2 can be used. Again, this could be problematic for warm roofs with Ct < 1 according to ISO 4355, Annex D. In any case, the drift model for multi-pitched roofs in ISO 4355:2013, Annex B, chapter B.3 needs to be calibrated. Furthermore, the formulae given in ISO 4355:2013, chapter B.3 cannot be used for calculation without major changes because of printing mistakes and contradictions between them. The acceleration due to gravity g = 9,81 m/s2 is missing. Instead of the density by mass r in kg/m3 the “weight” density r · g in kN/m3 is to be used. Figure B.4 is incomplete and contradictory. It does define the width “W” of the roof trough and the width “w” of the snow in the trough after sliding and the roof angles b1 and b2, but it does not define the heights h and d. However, as in other parts of the standard, h seems to be the height of the roof as the obstacle and d seems to be a snow depth. However, which snow depth? The snow depth “d” could be the equivalent roof snow depth after sliding distributed uniformly over the width w. Some of the formulae seem to support this. However, more likely “d” is the maximum roof snow depth in the trough after sliding, as it is used in other parts of the document. In this case, there are some mayor corrections required. ISO 4355:2013, Figure B.4 - “Snow load distribution – multi-pitched roof” shows a trough half filled with snow after complete sliding. This would be the slide load case 2. This load case is shown in the

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figure as a steep triangle over the reduced width w of the snow in the trough. This part of the figure is in contradiction to the snow load distribution shown above it, which belongs to load case 1.

Given are the following formulae: ISO 4355:2013, Annex B, chapter B.3 Multi-pitched roofs

Basic load case: sb = 0,8 s0 · Ce · Ct · µb (B.5)

Sliding load case 1: h < (0,8 s0 · Ce · Ct · µb · W / (r · (tan (90°- b1) + tan (90°- b2))))1/2 (B.6) d = (0,4 s0 · Ce · Ct · µb / r) + h/2 (B.7) ss1min = 2 r ((0,4 s0 · Ce · Ct · µb / r) - h/2) (B.8) ss1max = 2 · r · d (B.9)

Sliding load case 2: How is “w” to be calculated? h ≥ (0,8 s0 · Ce · Ct · µb · W / (r · (tan (90°- b1) + tan (90°- b2))))1/2 (B.10) d = (0,8 s0 · Ce · Ct · µb · W / (r · (tan (90°- b1) + tan (90°- b2))))1/2 (B.11) ss2min = 0 (B.12) ss2max = 2 · r · d (B.13)

where: d and h are not defined sb is the basic roof snow load in kN/m2

s0 is the characteristic ground snow load in kN/m2

Ce is the exposure coefficient

Ct is the thermal coefficient

µb is the basic shape coefficient

b1 and b2 are the roof angles of the two pitches in degrees

r is the snow density in kg/m3 in the trough after sliding (r = 300 kg/m3 is suggested)

W is the width of the roof trough in m

w is the width of the snow in the trough after sliding in m Furthermore, the given equations are wrong. For load case 2 equation (B.11) gives the snow depth “d” of the basic load case (B.5) assuming the snow density r after sliding is corrected by multiplying it with g = 9,81 m/s2. Two times this value above the gutter (B.13) and zero above the ridge (B.12) describe a triangular snow load distribution over the entire width of the trough W, no complete sliding as long as the snow does not fill the trough completely. Only in this case the equations would be correct. However, in this case the figure would be misleading. The selection criterion (B.6) and (B.10) compares the height of the trough “h” with the snow depth of the basic load case (B.5) using r · g. However, if “h” and depth of the uniformly distributed basic snow load would be equal, the snow would not fit in the triangular trough. Half of it would not fit in and be left on top, as in load case 1. Therefore, this cannot be the criterion to separate both load cases from each other. Conclusion: This model cannot be used without basic corrections. In the following corrections it is assumed that ”d” is the maximum depth of the snow in the trough after sliding. Slide load case 2 (trough is not full) is corrected first, because slide load case 1 (trough is full) is based on it.

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Correction: Slide load case 2: Snow height smaller than roof height

The triangular load after sliding Ss in kN/m roof length can be written as: Ss = r · g · d/2 · w. The roof snow load before sliding can be written as: SR = 0,8 · s0 · Ce · Ct · W. The snow load is to be redistributed and applying SR = Ss results in the following: 0,8 · s0 · Ce · Ct · W = r · g · d/2 · w d = 1,6 · s0 · Ce · Ct · W / (r · g · w)

For the geometry of the roof trough and the slide load in the trough the following geometric ratios apply: tan (90°- b1) = cot b1 = W1/h = w1/d tan (90°- b2) = cot b2 = W2/h = w2/d Roof width: W = W1 + W2 = h · (tan (90°- b1) + tan (90°- b2)) = h · (cot b1 + cot b2) Snow width: w = w1 + w2 = d · (tan (90°- b1) + tan (90°- b2)) = d · (cot b1 + cot b2)

The depth of the snow after sliding can be written as: d = 1,6 · s0 · Ce · Ct · h / (r · g · d) d2 = 1,6 · s0 · Ce · Ct · h / (r · g) d = (1,6 · s0 · Ce · Ct · h / (r · g))1/2

The roof height h can be replaced by the roof width W by introducing the roof angles, but this complicates the calculation as follows: d = (1,6 · s0 · Ce · Ct · W / (r · g · (cot b1 + cot b2)))1/2 d = (1,6 · s0 · Ce · Ct · W / (r · g · (tan (90°- b1) + tan (90°- b2))))1/2

Because the snow depth in the trough has the height d, not 2 d, the condition for the differentiation of slide load case 1 and 2 is also to be corrected: h < / ≥ d = (1,6 · s0 · Ce · Ct · h / (r · g))1/2 h < / ≥ d = (1,6 · s0 · Ce · Ct · W / (r · g · (cot b1 + cot b2)))1/2 h < / ≥ d = (1,6 · s0 · Ce · Ct · W / (r · g · (tan (90°- b1) + tan (90°- b2))))1/2

For the design with load case 2 the width w of the snow in the trough has to be estimated using the trigonometrically relations to the snow depth and the roof angles: w = d · (tan (90°- b1) + tan (90°- b2)) = d · (cot b1 + cot b2)

The snow load values are: ss2,min = 0 and ss2,max = r · g · d

Slide load case 1: Snow height larger than roof height.

The snow load consists of a triangular part filling the trough and a rectangular rest above the trough. The roof snow load before sliding can be written as: SR = 0,8 · s0 · Ce · Ct · W. The roof snow load after sliding Ss in kN/m roof length can be written as: Triangular part in the trough: Ss,T = r · g · h/2 · W. Rectangular part above the trough: Ss,R = r · g · Dd · W.

The snow load is to be redistributed and applying SR = Ss,T + Ss,R results in the following: 0,8 · s0 · Ce · Ct · W = r · g · (h/2 + Dd) · W h/2 + Dd = 0,8 · s0 · Ce · Ct / (r · g) Dd = 0,8 · s0 · Ce · Ct / (r · g) - h/2

The maximum height d of the snow in the middle of the trough can be corrected as follows: d = h + Dd = h/2 + 0,8 · s0 · Ce · Ct / (r · g)

As a result, the snow load values for sliding and redistribution can be corrected as follows: Above the ridge: ss1,min = r · g · Dd = 0,8 · s0 · Ce · Ct - h/2 · r · g Above the gutter: ss1,max = r · g · d = 0,8 · s0 · Ce · Ct + h/2 · r · g

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Despite the open questions with reference to the drift losses in comparison to flat roofs and the required major corrections, the snow load redistribution model for multi-pitch roofs in ISO 4355:2013 is more realistic than any other model so far, compare Figure 1 and Figure 2. However, it is questionable whether the described elaborate approach would be necessary for small greenhouse roofs. The acceptance because of so many corrections and the danger of further printing mistakes could be problematic. For greenhouses covered with glass or plastic film two architectural types are common. Type Venlo has gable roofs with a span between 3,2 m and 5 m for glass and up to 6,4 m for plastic film. The roofs are between 0,6 m and 1,3 m high and carried by horizontal lattice girders, see also Figure 2. American type greenhouses have larger roofs carried directly by a steel structure. On the other hand, the snow heights on the roof remain small when controlled heating with internal temperatures of 12°C or 18°C is used. That is why the slide load case 2 has more importance for greenhouses than the slide load case 1. However, case 1 is possible too for unheated roofs (Ct = 1) and larger snowfall rates in colder climate. 9.3.3 The incomplete sliding model in ASCE 7-10 An incomplete redistribution model for roof snow on multi-span roofs has been published by the American NGMA (National Greenhouse Manufacturers Association, Design Load Standard, 2004). The regulation is applicable for “Multiple Folded Plate, Sawtooth and Barrel Vault Roofs” alike. The model has 50% of the roof snow remaining at the ridge (or crown for vaulted roofs) and a maximum in the trough or gutter (3 pf / Ce) of three times the roof snow load pf divided by the exposure coefficient Ce. A similar model can be found in ASCE 7-10 (see Table 7) but with a smaller factor of 2 pf / Ce in the trough. These factors seem still large, but in the USA the roof snow is calculated with a general conversion factor of 0,7 only. This can be compensated by thermal factors Ct between 1,1 and 1,3 for cold roofs; as opposed to Ct = 1, which requires a warm roof on a closed building with qi >> 0°C. The influence of melting on the roof snow loads is slightly more dominant than the influence of drifting. Therefore, single values of Ce or Ct cannot be compared directly to the appropriate values in the Eurocode, because the ranges are different. The total roof snow load should be compared, as can be seen in Table 7. The incomplete redistribution model has two load cases, balanced loads and unbalanced loads. The balanced load case 1 replicates the flat roof snow load. This load case would be used to design structural parts at the crown. The unbalanced load case 2 gives a local maximum for the gutter and also the maximum total load for the structure. The important findings for load case 2 for multi-span roofs are, how precipitations are redistributed from the ridges down into the trough. However, the unbalanced load case 2 is realistic for certain ratios of roof snow height to roof height and for cold rough cladding. For high roofs with slippery cladding in combination with small snow loads or a low thermal coefficient (warm roof) a complete sliding is very likely. In this case the load in the valley is too small. The load at the ridges can also be smaller / larger than 0,5 pf.

25

Multi-span pitched / arched roof ASCE 7-10, chapter 7.4.4, 7.6.3 and Figure 7-4 Balanced Load Case 1:

Flat roof snow load: pf = 0,7 · Ce · Ct · pg (for Importance Factor Is = 1 and Roof Slope Factor Cs = 1) with: Ce Exposure Factor, with 0,7 ≤ Ce ≤ 1,2, see classification in Table 8

Ct Thermal Factor, with 0,85 ≤ Ct ≤ 1,3, see classification in Table 8 pg Characteristic ground snow load, with 1,2 kN/m2 ≤ pg ≤ 14,4 kN/m2

Total snow load per trough: Ratio of conversion pf / pg Thermal Factor Ct

Exposure Factors Ce 1,2 1,1 1 0,9 0,8 0,7

1,3 1,092 1,001 0,91 0,819 0,728 0,637 1,2 1,008 0,924 0,84 0,756 0,672 0,564 1,1 0,924 0,847 0,77 0,693 0,616 0,517 1 0,84 0,77 0,7 0,83 0,56 0,4704

0,85 0,714 0,6545 0,595 0,5355 0,476 0,3998 Unbalanced Load Case 2:

Locally at the ridges: Min: ps1 = 0,5 pf Locally in the valley: Max: ps2 = 2 pf / Ce Total snow load: Ps = (ps1 + ps2) / 2 · W (with: W – total width of the trough)

Total snow load per trough: Ratio of conversion (Ps / W) / pg Thermal Factor Ct

Exposure Factors Ce 1,2 1,1 1 0,9 0,8 0,7

1,3 1,183 1,16025 1,1375 1,11475 1,092 1,069 1,2 1,092 1,071 1,05 1,092 1,008 0,987 1,1 1,001 0,98175 0,9625 0,94325 0,924 0,90475 1 0,91 0,8925 0,875 0,8575 0,84 0,8225

0,85 0,7735 0,7586 0,74375 0,7289 0,714 0,699

ASCE 7-10, chapter 7.3 Generalized information taken from ASCE 7-10, Table 7.2 and 7.3

Classification of thermal conditions and terrain category / climate / roof exposure Thermal Factors

Ct

1,3 Structures intentionally kept below freezing 1,2 Unheated open and open-air structures

1,1 Structures kept just above freezing; ventilated roofs only if U < 0,23 W/m2K between heated and ventilated space

1 Other structures 0,85 Continuously heated greenhouses (qi > 10°C; roof with U > 2,5 W/m2K; safety precautions)

Exposure Factors

Ce

1,2 Warm / calm climate, sheltered roof 1) 1,1 Moderate climate, sheltered roof 1) 1 Cold climate, sheltered roof 1); warm / calm and moderate climate, partially exposed roof

0,9 Warm / calm + moderate climate, fully exposed roof 2), cold climate, partially exposed roof 0,8 Cold climate, fully exposed roof 2); very cold climate / terrain 3), partially exposed roof 0,7 Very cold climate / terrain 3), fully exposed roof 2)

1) Sheltered roof: by terrain, higher structures or trees present during the expected working life of the structure 2) Fully exposed roof: no shelter on all sides, no parapets higher than the expected roof snow, no roof obstructions 3) Very cold climate / terrain: Alaska, if site with no trees within 3 km radius; roof above the treeline in windswept mountainous areas

Table 7: Snow redistribution model for multi-span roofs according to ASCE 7-10

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Notes and Comments to Table 7:

1. The general roof-ground conversion factor ks = 0,7, defined as reference standard in the USA cannot be compensated by the exposure factor maximum of Ce = 1,2 for sheltered roofs in warm climate alone (0,7 · 1,2 = 0,84). To reach the ground snow level on a roof, thermal factors Ct > 1 are required too (on unheated buildings, open air structures and freezer buildings, ice-skating halls). That means in reverse that large roof snow reductions are rather the result of melting than of drift losses. The Eurocode (ks = 0,8 included in the shape coefficient, 0,8 ≤ Ce ≤ 1,2 for the exposure of the roof, Ct = 1) has not covered this point. ISO 4355:2013 is also not clear about it.

2. The largest ratio of roof to ground snow can be calculated using Ce = 1,2 and Ct = 1,3 giving 0,7 · 1,2 · 1,3 = 1,092 in load case 1. This is only 9% above the ground snow level and can be explained due to refreezing / resublimation. 18% in load case 2 raises questions. Total roof snow loads in excess of the ground snow load as for Ct = 1,3 are only possible locally and at the roof edges (first valley, windward side). For the global loads of large roofs, the ground snow load is the limit.

3. It is doubtful, whether the exposure factor Ce can have a large influence on the reduced snow loads on greenhouses, because the presence of roof snow during rapid melting is very short. However, if Ct is kept constant at Ct = 0,85 and if lower Ce correlate to larger ground snow loads, the sk-dependence of Ct is covered by this regulation. ISO 4355, Annex D gives a thermal coefficient with a dependence on sk. That Ct uses Ce = 1 and µ = 0,8.

4. For unbalanced loads, the limited variation of roof snow loads for different Ce is a result of the drift limitation for snow in the valley. This should also be applied to balanced loads.

5. Because the model works with constant ratios of roof to ground snow load and does not check whether the trough can accommodate the snow, it fails for very large ground snow loads in comparison to the trough size.

9.4 Conclusions and greenhouse model EN 13031-1:2019 Multi-span greenhouse roofs are often very large with many small roof troughs. However, for heated greenhouses a correction for the influence of the roof size is not required. The thermal coefficient Ct is based on wind-corrected precipitation data, see Part I of this Background report. The Ct-value has been calibrated, to be used together with a standard shape factor for flat roofs of µ1 = 0,8. In ISO 4355:2013 instead of µ1 = 0,8 a general conversion factor of 0,8 is used; in ISO 4355:1998 a general value of Ce = 0,8. In all three cases the ground snow load level is already covered when using Ct < 1. If the thermal coefficient Ct is estimated using thermodynamic models and precipitation data, it has to be kept in mind, that no drift losses are possible in the very short time it takes to remove the snow by melting. Therefore, Ce = 1,25 would be appropriate and the roof size is without influence. From these findings the following recommendations can be given in prEN 13031-1. For heated greenhouse roofs with controlled heating (Cm = 1,333) the thermal coefficient of Ct < 1 according to ISO 4355, Annex D (Sandvik formula) is used. It accounts for the full precipitation despite using µ1 = 0,8. Therefore the redistributed snow accumulation above inner gutters is µ2 = 2 · 0,8 = 1,6 and above the ridge µ0 = 0.

For greenhouse roofs without controlled heating (Ct = 1 and Cm = 1,2+), but kept well above freezing with internal air temperatures qi ≥ 5°C, a certain amount of snow will become molten before sliding completely. Therefore, the roof snow for average conditions (assuming 20% melt losses) can be redistributed with µ1 = 0,8 and Ce = 1, giving µ2 = 1,6. For other Ce values, the maximum shape coefficient above the inner gutter is to be corrected µ2 = 1,6 / Ce. Because of immediate sliding on warm roofs, drift losses can have no influence.

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Figure 3 (Photo VDH): Sliding and melting after snow fall event November 2010 The complete sliding of snow into the trough as shown in Figure 3 can be seen on all other case studies for heated greenhouses with transparent cladding. Many unusual field observations were possible in the winter of 2010. Snowfall events as early as November made it possible to observe the accumulation and sliding of snow on cold greenhouse roofs, because many heating devices were not operative yet. For unheated roofs, the influence of melting is further limited. It is possible, that the snow does not slide completely. However, as a limit state and for simplification complete sliding is assumed with µ0 = 0 on the ridges and the maximum value of µ2 above the gutter. The total snow load results from the snowfall into the lower part of the trough (ca. 50%) without drift losses plus the sliding snow from the upper part (50%) near the ridges, where at least for small roofs in colder climate drift losses are possible. From these drift losses 60% are redistributed onto the same roof. These ratios have been derived from the comparison with the incomplete redistribution model in ASCE 7-10 adapted to EN-format (new: prEN 1991-1-3:2020) as follows. An overview on all results is given in Table 9. Flat roof snow load: si = µ1 · Ct · sk = 0,8 · Ce · Ct · sk with: µ1 = 0,8 Ce for small flat roofs where: Ce is the exposure coefficient with 0,8 ≤ Ce ≤ 1,2 and Ce = 1,25 for large roofs Ct is the thermal coefficient, with 0,2 ≤ Ct ≤ 1,2 sk is the characteristic ground snow load in kN/m2 µ1 is the shape coefficient (includes general conversion ks = 0,8) Redistribution into the trough: Snow load S in kN/m with W - total width of the trough Bottom of the trough (50%): precipitation - melting: S = Ct · sk · W / 2 From the ridge area (50% = 1/2) with drift losses: S = 0,8 · Ce · Ct · sk · W / 2 60% = 3/5 of the drift losses from the ridges: S = (1 - 0,8 Ce) · Ct · sk · W / 2 · 3/5

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Total snow load in the trough: S S/W = sk · Ct · (½ + ½ · 0,8 Ce + ½ · (1 - 0,8 Ce) · 3/5) S S/W = sk · Ct · (0,8 + 0,16 Ce) Shape coefficients for a simple triangular distribution: Above the ridge: µ0 = 0 Above the gutter: µ2 = 2 · S S/W / (sk · Ct) = 1,6 + 0,32 Ce For roofs of permanently open buildings, porches or canopies, or for cold, ventilated roofs the ground snow load level can be expected in the trough. For roofs above closed buildings, where the temperature will not be colder than the environment, a limited melting (4%) can be taken into account. The snow load distributions as proposed for EN 13031-1:2019 are summarized in Table 8.

Load Case 1 Uniform distribution

Roof pitch angle a in ° a < 60°/Cm a ³ 60°/Cm

Outer roof pitches µ1 = 0,8 Ce · (cos (1,5 Cm · a) )0,5 µ1 = 0

Inner roof pitches Independent of the roof angle: µ1 = 0,8 Ce

Load Case 2 Sliding, Drifting and Accumulation

Outer roof pitches Windward Side: µLuv = 0 Leeward Side: µLee = µ1 or 0 2)

Inner roof pitches Above the Ridge µ0 = 0

heated warm

Ct < 1; Cm = 1,333 Ct = 1; Cm = 1,2+ Above the inner Gutter µ2 = 1,6 1)

not heated cold

Closed building; Cm = 1,2 Above the inner Gutter µ2 = 1,54 + 0,308 Ce

Open, exposed, cold ventilated roof; Cm = 1 Above the inner Gutter µ2 = 1,6 + 0,32 Ce

Where: Cm is the surface material coefficient Ce is the exposure coefficient with 0,8 ≤ Ce ≤ 1,2; for large roofs Ce = 1,25 1) Ce = 1, because Ct is based on full precipitations, but calibrated with Ce = 1 and µ1 = 0,8 for flat roofs. 2) Both versions (before and after sliding) can be applied, if required.

Table 8: Shape Coefficients and Snow Load Distribution for multi-span pitched roofs according to EN 13031-1:2019, Table C.4 and C.5

Type of

Greenhouse Surface Material

Coefficient Thermal

Coefficient Exposure Coefficient Ce

1,25 1,2 1 0,8

unheated Cm = 1 Ct = 1 1

(µ2 = 1,6) 0,992

(µ2 = 1,653) 0,96

(µ2 = 1,92) 0,928

(µ2 = 2,32)

Cm = 1,2 Ct = 1 (0,9625 incl.)

0,962 (µ2 = 1,54)

0,9548 (µ2 = 1,591)

0,924 (µ2 = 1,848)

0,8932 (µ2 = 2,233)

heated Cm = 1,2+ Ct = 1+

(0,8 incl.) 0,8

(µ2 = 1,28) (µ2 = 1,333) (µ2 = 1,6) (µ2 = 2)

Cm = 1,333 Ct < 1 0,8 Ct (for Ce = 1: µ2 = 1,6)

Table 9: Total snow load (S S/W)/sk and shape coefficient µ2 for small multi-span greenhouses In exceptional cases of snow heights larger than the height of the trough, the snow load distribution needs to be corrected. The shape coefficient µ0 above the ridge becomes larger than zero and the shape coefficient µ2 above the gutter becomes somewhat smaller. This is why for the gutter it remains on the safe side, if the correction is forgotten.

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Figure 4: (Figure made by TNO): Snow load distribution on pitched roofs of heated

greenhouses with Ct < 1 and Cm = 1,333 or for Ct = 1 and Cm = 1,2+ The triangular shape of the snow accumulation in load case 2 for multi-span roofs is limited to the depth of the trough hr. This can be controlled as follows: s2,n,t = sk,n · Ct · µ2 ≤ hr · gs where s2,n,t is the roof snow load above the inner gutter in kN/m2 where µ2 is the shape coefficient above the gutter according to Table 8 hr is the depth of the trough or the roof height in m gs is the equivalent (force) density of the sliding snow, with: gs = 3,5 kN/m3 for ripe snow on warm roofs during melting (Ct < 1) and gs = 2 kN/m3 for drifted and settled fresh snow on cold roofs (Ct = 1) If the volume of the snow is larger than the trough volume (s2,n,t > hr · gs), the excess snow is distributed uniformly. The snow load distribution can be corrected as follows: Above the ridge: s0,n,t = ½ · (sk,n · Ct · µ2 - hr · gs) µ0,cor = ½ · (µ2 - (hr · gs)/(sk,n · Ct)) Above the gutter: s2,n,t = s0,n,t + hr · gs µ2,cor = ½ · (µ2 + (hr · gs)/(sk,n · Ct))

Figure 5: (Figure made by TNO) Non-uniform snow load distribution for large height ratios

snow / roof

b) Multi pitched roofs

µ2 µ2µ1µ1

µ1µ11

2

a) Duo pitched roofs

µ1

µ11

2

Θi ≥ 5°C Θi ≥ 5°C

µ1 µ1

Multi pitched roofs

hr

µ0,corµ2,cor

µ0,corµ2,cor

µ0,cor

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9.5 General redistribution model for multi-span roofs - Proposal A proposal for a general redistribution model for multi-span roofs is given in Table 11. Load case (i) is based on the bi-linear roof angle function for the basic shape coefficient µ2,b. In load case (ii) the shape coefficient µ2 is used for the ridge and µ3 is used for the gutter. ISO 4355:2013 shows this snow load distribution in Figure B.4. Because of the shortcomings of Figure B.4 as discussed in chapter 9.3.2, the geometry and snow load distribution for the roof trough is given in Figure 6.

Figure 6: Geometry and unbalanced snow load distribution load case (ii) in roof troughs

For the internal troughed roof area a special exposure coefficient Ce,T is defined. As for flat roofs the special exposure coefficient for troughs depends on the roof size. In this case this can be defined by the number of troughs per roof. As in other cases the reference value for a = 0° (flat roof) is 0,8.

Ce,T=1,25-n·(1,25-Ce)Where Ce is the exposure coefficient for the roof location

n is the drift loss part from the trough (Flat roof reference value: 1 - 0,8 Ce)

From small roofs with one or two troughs drift losses are possible, as the measurement in Ottawa 1971 has shown with Cb · Ce,T = 0,75. For a flat roof at the same location on the airport according to NBCC the flat roof snow load level would have been Cb · Cw = 0,8 · 0,75 = 0,6. In ASCE 7-10 it is recommended to allow half of the flat roof drift losses from the ridges. The other half is distributed back into the trough. The drift loss part would be: n = ½. Using the special exposure coefficient Ce,T, the ratio of roof to ground snow load would be 0,8 Ce,T = 1 - ½ (1 - 0,8 Ce). For

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normal exposure Ce = 1 the special exposure for the troughed area would be 0,8 Ce,T = 0,9. This leads to 10% drift loss from a small roof trough in comparison to 20% from a small flat roof. Such a percentage is already indicated in prEN 1991-1-3:2020. For large roofs (flat roofs as reference: max Ce,F = 1,25) at the inner troughed roof area applies: 0,8 Ce,T = 0,8 · 1,25 = 1 (roof snow = ground snow). Special exposure coefficients for different troughs in comparison to the exposure coefficient on flat roofs are shown in Table 10.

Special Exposure Coefficient Trough Multi-pitched Roof Ce,T = 1,25 - n · (1,25 - Ce)

Location Drift Loss Potential Part n

Exposure Coefficient Ce 1,25 1,2 1 0,8

Internal roof area, large roof, many troughs

no drift loss (0%) n = 0 1,25 1,25 1,25 1,25

Small roof, single trough drift loss from the ridges (50%) n = 0,5 1,25 1,225 1,125 1,025

Flat roof (Comparison) 100% drift losses n = 1 1,25 1,2 1 0,8

Table 10: Special exposure coefficient for troughed roof area of multi-pitched roofs Note: Are drift losses as large as for a flat roof allowed as in ISO 4355:2013, the drift loss part is n = 1 and the exposure coefficient becomes Ce,T = Ce. The model is consistent with other models in ISO 4355. However, this is not recommended here. It does also not agree well with the measurements. On the outer roof pitches on the windward and the leeward side of a multi-span roof, the same rules as for other sloped roof surfaces apply. However, there are no saddle-roof drift surcharges across multi-span roofs. The drifting is included in Ce,T > Ce. For a consistent model as shown in Table 11 for the unbalanced Load Case (ii) for large and small troughs and large and low snow loads, two cases need to be separated, Case 1 and Case 2. Is the volume of the snow smaller than the trough volume, the distribution of the snow within the trough is triangular (Case 1). Is the volume of the snow larger than the trough volume, in addition to the triangular distribution in the full trough, there is a rectangular distribution of the access snow on top of the trough (Case 2). Which case applies can only be found out by checking. For the check in the first step it is assumed the snow volume is smaller than the trough volume. The snow height of the assumed triangular distribution d1 = (sk · Ct · Ce,T · 1,6 h / g)1/2 is calculated and compared with the height of the trough h. In Case 1 the assumed snow height d1 is smaller than the trough height h (h ≥ d1). Then the assumption about the snow load distribution is correct and the maximum snow height above the gutter is d = d1. Using the geometric relations between triangular trough and triangular snow distribution within the trough, in a second step the reduced width w of the snow cover in the trough can be calculated with w = w1 + w2 = d (cot a1 + cot a2). For very small snow loads in large troughs, the shape coefficient maximum can become larger than µ3 = 2, however together with a very small snow-covered width w in the trough. The designer can decide how to distribute the snow across the structural element to be calculated, e.g. a glass plate. This is the designer’s responsibility and needs no further guidance. In Case 2 the assumed snow height d1 is larger than h (h < d1). Then the assumption of a triangular distribution is not correct. The distribution consists of a triangle with the trough height h and the width W of the trough. Above ridge level there is the rectangle across the full trough width W with

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the unknown height Dd above the ridge. In Case 2 in the second step the correct snow height d = h + Dd needs to be calculated.

Load Case (i)

Balanced Load Arrangement

Shape Coefficient µ2 (a) for uniform distribution Roof angle in ° Windward Inner troughed area Leeward

5° < a ≤ 30° µ2,b = 0,8 Ce

µ2 = 0,8 Ce,T

µ2,b = 0,8 Ce

30° < a < 70° µ2,b = 0,8 Ce (70° - a1) / 40° µ2,b = 0,8 Ce (70° - a2) / 40°

a ³ 70° µ2,b = 0 µ2,b = 0

Format Roof Snow Load: s = sk · Ct · µ2

Load Case (ii)

Sliding +

Drift

Unbalanced Load Arrangement

Wind- and

leeward 1)

For the inner troughed area

Check: h ≥ / < d1 = (sk · Ct · Ce,T · 1,6 h / g)1/2

µ2 = 0 or

µ2 = µ2,b

h ≥ d1: Case 1 h < d1: Case 2

d = d1 d = h + Dd with: Dd = 0,8 Ce · sk · Ct /g - h/2 · Ce/Ce,T

w = w1 + w2 w = d (cot a1 + cot a2)

w = w1 + w2 = W1 + W2 w = W

Above the ridge

smin = 0 smin = g · Dd

smin = 0,8 Ce · sk · Ct - h/2 · g · Ce/Ce,T

µ2 = 0 µ2 = 0,8 Ce (1 - g · h/(1,6 Ce,T · sk · Ct))

Above the

gutter

smax = g · d smax = g · (h + Dd) smax = (1,6 h · g · sk · Ct · Ce,T)1/2 smax = 0,8 Ce · sk · Ct + g · h (1- ½ Ce/Ce,T)

µ3 = ((1,6 h · g · Ce,T)/(sk · Ct))1/2 µ3 = 0,8 Ce + g · h/(sk · Ct) · (1- ½ Ce/Ce,T)

The exposure coefficient in the troughed area is Ce,T = 1,25 - n · (1,25 - Ce) Recommendation: Internal troughed area of large roofs: no losses: n = 0: Ce,T = 1,25 Small roof, one trough: Losses from the ridges only: n = 0,5; Ce,T = 1,25 - ½ (1,25 - Ce)

Format Roof Snow Load: s = smin … smax or: s = sk · Ct · µi with i = 2 and 3 Ce Ce,T n Ct a1, a2 h W d w g

exposure coefficient, with 0,8 ≤ Ce ≤ 1,2; special exposure coefficient for troughed roofs, with Ce,T ³ Ce; drift loss part from the trough (NDP; suggested is n = 0 (no drift loss) for large roofs with many troughs; otherwise n = 0,5 (drift loss from the ridges, not from the lower part of the trough (50% of the area); thermal coefficient Ct > 1 for very cold roofs; roof angles a in ° for the two roof surfaces of the trough i = 1 und i = 2; trough height (mean value, usually equal to the roof height) in m; total width of the trough in m (W = W1 + W2); maximum snow height in the trough in m; width of the snow cover in the trough in m (w = w1 + w2); equivalent snow weight density in the trough (NDP; suggested is g = 2 kN/m3 for cold roofs, for warm roofs rather g = 3 kN/m3 to 3,5 kN/m3.)

1) For outer roof surfaces in load case (ii) the snow load distribution should consider sliding Table 11: Proposal for a general snow load distribution for multi-span roofs – in the format

according to prEN 1991-1-3:2020

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Remark to the Format: Table 11 is large and seems complicated, because it contains three alternative methods for load case (ii). Shape coefficients µi, snow load formulae and most simple relations for the local snow loads smin above the ridge and smax above the gutter can be calculated. It becomes clear, that the calculation with simple relations for smin and smax is the easiest. The risk for printing mistakes is smaller too. Calibration and comparison: The calibration of the proposed model can be seen in Table 12. The results are calculated for a thermal coefficient of Ct = 1. For Ct = 1,2 the results would be larger by a factor of 1,2. This would not be correct. However, this mistake cannot be repaired within this model. The value is already much too large for a flat roof. Thermal coefficients Ct > 1 have to be calibrated, see ASCE 7-10.

Load Case (i): Balanced Load arrangement Flat roof: s1,F = µ1,F · Ct · sk (Shape coeff. µ1,F = 0,8 · Ce) Exposure coefficient: Ce see below Trough: s2 = µ2 · Ct · sk (Shape coefficient µ2 = 0,8 · Ce,T) Exposure coefficient trough: Ce,T = 1,25 - n · (1,25 - Ce)

Total Snow Load in the Trough: Ratio s2 / (sk · Ct) for Load Case 1 1) Drift Loss

Part n Exposure Coefficient Ce

1,25 1,2 1 0,8 Internal roof area, large roof n = 0 1 1 1 1

Single trough n = 0,5 1 0,98 0,9 0,82 Flat roof (Comparison) n = 1 1 0,96 0,8 0,64

Load Case (ii): Unbalanced Load Arrangement

Case 1 (h ≥ d1) Case 2 (h < d1) Local, above Ridge: s2 in kN/m2 s2 = 0 s2 = smin = g · Dd Local, above Gutter: s3 in kN/m2 s3 = smax = g · d s3 = smax = g · (h + Dd)

Snow Height d in m d = d1 d = (sk · Ct · Ce,T · 1,6 h / g)1/2

Dd = 0,8 Ce · sk · Ct / g - h/2 · Ce/Ce,T d = h + Dd

Width of the Snow Cover w in m w = d · 2 cot a = d · W/h w = W (Width of the Trough in m) Total Snow Load in the Trough in

kN/m S ST = smax/2 · w = d · g · w / 2 S ST = 0,8 Ce,T · sk · Ct · W

S ST = (smin+smax)/2 · W = g (Dd + h/2) · W S ST = (0,8 Ce · sk · Ct + g · h/2 · (1- Ce/Ce,T))· W

Total Snow Load in the Trough: Ratio (S ST / W) / (sk · Ct) for Load Case 1 1)

Drift Loss Part n

Exposure Coefficient Ce 1,25 1,2 1 0,8

n = 0 (Ce,T = 1,25) 1 1 1 1 n = 0,5 (Ce,T = 0,625 + Ce/2) 1 0,98 0,9 0,82

Classification of the thermal conditions and the Exposure of the Site / Winter Climate:

Thermal Coefficient

Ct

1,2 Buildings, intentionally kept below zero (freezer buildings, ice skating arenas) 1 Other structures

<< 1 Greenhouses with transparent cladding and controlled heating (qi >> 5°C; cladding: Uo > 1 W/m2K; further safety requirements see EN 13031-1:2019

Exposure Coefficient

Ce

1,25 Ce,F: Flat roof – Limit for large roof size Lc ³ 400 m 1,2 sheltered site, e.g. Terrain category 4 in DIN EN 1991-1-4 1 normal, e.g. Terrain category 3 in DIN EN 1991-1-4

0,8 windswept (4,5 m/s), exposed site, e.g. Terrain category 0,1 oder 2 in DIN EN 1991-1-4 1) In load case 2 the total snow load depends also on the ratio of snow height above the trough Dd and trough height h.

Table 12: Results for the total Snow Load – General Snow Load Distribution Model

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Furthermore, if the proposed model is used for glass roofs together with a thermal coefficient Ct << 1, the roof snow loads are short-term loads due to precipitations of few hours to three days. Drift losses should not be considered (Ce = 1,2). However, if the thermal coefficient Ct according to ISO 4355, Annex D is used, Ce,T = Ce = 1 can be applied. This thermal Coefficient Ct is developed on the basis of wind-corrected precipitations and calibrated to match a roof to ground snow conversion of ks = 0,8 (by factor, shape or exposure coefficient), see Part I of this Background report. Table 12 shows, that the total snow loads above valleys or troughs, from where it is difficult to remove snow, do not depend on the roof angle, except for very small roof angles in transition to the flat roof. In this proposal the load cases (i) and (ii) are consistent, as it should be. Not shown are results for Load Case 2, because total snow load depends additionally on the height of the snow above the ridge level Dd in comparison to the trough height h. For the part Dd instead of the exposure coefficient of the trough Ce,T, the exposure coefficient of the flat roof surface Ce can be used. The larger this height Dd becomes, the more the total roof snow load will tend towards the flat roof snow load. For small snow loads and large roof troughs Load Case 1 is relevant, for large snow loads and small troughs Load Case 2. Generally, the limit between Load Case 1 und 2 can be estimated for a ratio of though height to snow load (h/sk)lim with the dimension of m3/kN:

(h/sk)lim = Ct · (1 - n · (1 - 0,8 Ce)) · 2/g Where h is the maximum trough height (depth) in m

sk is the characteristic ground snow load in kN/m2 Ce is the exposure coefficient for the roof location with 0,8 ≤ Ce ≤ 1,25 Ct is the thermal coefficient g is the equivalent (force) density of the snow in kN/m3 n is the drift loss part from the trough (flat roof reference value: n = 1; single

trough, small roof: n = 0,5; internal roof area, large roof, many troughs: n = 0)

For equivalent snow densities of g = 2 kN/m3 and standard conditions with Ct = Ce = 1 the limit ratio is (h/sk)lim = 1 - 0,2 · n

The snow load distribution on the airport roof in Ottawa, Canada, during the Blizzard of the Century 1971 according to Figure 1 and Table 3 can be approximated with the proposed model. The Canadian standard NBCC is taken into account with the basic shape coefficient Cb = 0,8 and the exposure coefficient for a very windy and exposed airport site with Cw = 0,75. For a flat roof as a reference a roof-to-ground snow load conversion of Cb · Cw = 0,8 · 0,75 = 0,6 would result, comparable to a value of 0,64 according to the Eurocode with µ1 = 0,8 Ce and Ce = 0,8. As ground snow load the measured value of s = 3,35 kN/m2 (based on g = 2,943 kN/m3) is taken into account, because this value is higher as the recently known characteristic ground snow load value for the Ottawa Macdonald-Cartier Airport of sk,50 = 2,8 kN/m2.

Note: The lower value is based on a set of more recent measurement data from 1983 - 2011, see Brooks, et.al. (2014). From this publication a characteristic flat roof snow load of s1,50 = 1,904 kN/m2 is known, giving a roof-to-ground snow load conversion of 0,68 > 0,6 according to NBCC. The lower values with less drift loss might be due to a milder and wetter climate.

For the two exposed troughs with a small height of h = 1,83 m (a = 18,5°), a medium value for the drift losses of n = 0,5 is assumed, giving an exposure coefficient for the trough of Ce,T = 1,25 - n · (1,25 - Ce) = 1,25 - 0,5 · (1,25 - 0,75) = 1 with Ce = Cw = 0,75 for the small flat roof according to NBCC.

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In a first step a triangular snow load distribution is assumed. The maximum snow height of this distribution would be d1 = (sk · Ct · Ce,T · 1,6 h / g)1/2 = (3,35 · 1 · 1 · 1,6 · 1,83 / 2,943)1/2 = 1,82563 m. This height is nearly as high as the trough h. The test gives: d1 = 1,8256 m ~ 1,83 m = h. The distribution is correct and the Load Arrangement Case 1 with d = d1 applies. In a second step the width w of the snow load distribution is calculated. In this case it is nearly equal to the width of the trough w = W = 11 m. With this information the snow loads can be calculated: Balanced Load Case (i):

Windward / leeward: Ce = 0,75: s2,b = µ2,b · Ct · sk = 0,8 · 0,75 · 1 · 3,35 = 2,01 kN/m2 Inner trough surfaces: Ce,T = 1: s2,b = µ2,T · Ct · sk = 0,8 · 1 · 1 · 3,35 = 2,68 kN/m2

Unbalanced Load Case (ii):

Above the ridge: smin = 0 (µ2 = 0) Above the gutter: smax = g · d = 2,943 · 1,83 = 5,3857 kN/m2 (µ3 = 1,60767)

The local value above the gutter meets the maximum measurement value of 5,428 kN/m2 in the first valley very well (< 1 % difference = with precision). The measurement value in the second valley is lower. The measured snow loads on the ridges are not zero as in the drift model for the unbalanced load case (ii). However, the model also has a balanced load case (i), which covers these and higher local loads above the ridges in case of snowfall conditions with less wind. The snow loads on the roof spans and on the roof in total are also met; 29,668 kN/m (80,5% of the ground snow) on the first span, 27,938 kN/m (75,8% of the ground snow) on the second span and 25,367 kN/m (68,8% of the ground snow) on the third span, resulting in 82,937 kN/m (75% of the ground snow) in total. The model gives 25,866 kN/m2 (70% of the ground snow) on the first and third spans due to Cw = 0,75 on the outer roof surfaces, but 29,62 kN/m2 (80% of the ground snow) on the second span, resulting in 81,36 kN/m (73,6% of the ground snow) in total. This is a difference between measurements and model of less than 2%. Only on the first leeward roof surface the measurement values show a larger drift than covered by the proposed model: 22,034 kN/m (119,6% of the ground snow). To simulate this, a drift surcharge from the windward roof surface would be required. As a result, the snow load in the first trough is also larger: 33,894 kN/m (91,98% of the ground snow). However, it has to be kept in mind, that the more recent evaluations imply less severe drift conditions for the Ottawa Airport (Cb · Cw = 0,8 · 0,85 = 0,68 > 0,6 = 0,8 · 0,75). Also, a more conservative choice of the drift loss part n < 0,5 can be made to increase the snow load in the first valley to 92%. At the Blizzard of the Century 1971, despite of the strong winds, the drift duration of three days for one single snowfall event was shorter than in usual winter seasons. Using Cw = 0,85, a smaller drift loss part of n = 0,2 for a single snowfall event would give Ce,T = 1,17 resulting in 34,49 kN/m (93,6% of the ground snow) as in the measurement. For a comparison with the model according to ASCE 7-10, the reference value of 0,7 has to be taken into account. Also, the total snow loads in the proposal according to the Eurocode cover all thermal conditions with Ct = 1, whereas ASCE 7-10 differentiates them with values Ct = 1 to Ct = 1,2. The Eurocode-Model can be placed in the middle of that range at about Ct = 1,1 according to ASCE 7-10. Furthermore, according to ASCE 7-10, 60% of the drift losses from the ridges (50% of the area) are redistributed back into the trough. Only 40% drift away from the roof. That refers to a drift loss part of n = 0,2 < 0,5. However, this value seems arbitrarily chosen, not suited to be used on all roofs. Instead, the already accepted approach according to prEN 1991-1-3:2020 is used with normal drift losses (1 - 0,8 Ce) from the upper part (ridges) and with n = 0,5 for single troughs only. For the internal troughs of large roofs n = 0 applies (no drift losses). However, the drift part n should be made a national choice (NDP) in the same way as the drift factor d for saddle roofs.

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Literature

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Devices – Solar Panels on Flat Roofs - Comments to ISO-4355-2013 Part I.pdf: The use of the thermal coefficient for large roofs - Comments to ISO-4355-2013 Part II.pdf: Critical review of the limits of the thermal coefficient - Comments to ISO-4355-2013 Part III.pdf: Influence of the roof angle on the thermal coefficient

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