CLT - Cross Laminated Timber Fire protection.

Size: px
Start display at page:

Download "CLT - Cross Laminated Timber Fire protection."

Transcription

1 CLT - Cross Laminaed Timber Fire proecion

2 C O N T E N T S Version 01/2016 AG I should be noed ha his fac shee on fire proecion is merely inended o suppor he user. Sora Enso Wood Producs does no assume any responsibiliy for he accuracy or compleeness of his documen. 1 Inroducion Fire proecion The building maerial, wood when exposed o fire Reacion-o-fire performance of building producs Fire resisance of building componens Classificaion Fire proecive cladding Fire resisance of CLT componens Verificaion of he fire resisance of CLT elemens based on classificaion repors according o EN CLT exernal wall srucures CLT wall srucures CLT ceiling srucures Verificaion of he fire resisance of CLT elemens based on calculaions according o EN :2011 (Eurocode 5) Verificaion mehod for acions in he fire siuaion according o EN : Verificaion mehod for mechanical resisance in he fire siuaion according o EN : Charring raes for Sora Enso CLT Design value of charring raes 0 for CLT on surfaces which are unproeced hroughou he duraion of he fire Design value of charring raes 0 for CLT on surfaces which are iniially proeced from exposure o fire by gypsum plaserboard Page 1 of 50

3 C O N T E N T S 5.4 Deermining he load-bearing capaciy (R) of CLT elemens according o EN : Deermining he inegriy (E) and insulaion (I) of CLT elemens The insulaing and proecive layers of a componen Deermining a layer s basic imes Calculaing he posiion coefficien k pos Deermining he join coefficien k j,i Calculaion model/applicaion for CLT Technical fire proecion of deailing Joins and connecions beween componens Insallaions and fixures Fire-rearding sealing in CLT consrucions Faseners Double-layered componens Bibliography Annex 1 Design examples A 1.1 Deermining he charring rae of Sora Enso CLT (non-faced) A 1.2 Deermining he charring rae of Sora Enso CLT (faced) A 1.3 Deermining he noional charring deph of Sora Enso CLT (non-faced) A 1.4 Deermining he inegriy (EI) of Sora Enso CLT (faced) Annex 2 Informaion on he residual cross-secion and inegriy of Sora Enso CLT componens A 2.1 Calculaed residual cross-secion and inegriy of CLT wall srucures (non-faced) A 2.2 Calculaed residual cross-secion and inegriy of CLT wall srucures wih single layered fire proecion plaserboard cladding (12.5 mm) on he fire-exposed side of he componen A 2.3 Calculaed residual cross-secion and inegriy of CLT wall srucures wih service caviy (50 mm, rock fibre) and single layered fire proecion plaserboard cladding (12.5 mm) on he fire-exposed side of he componen A 2.4 Calculaed residual cross-secion and inegriy of CLT ceiling srucures (non-faced) Page 2 of 50

4 I N T R O D U C T I O N 1 Inroducion 1.1 Fire proecion Fire proecion in is enirey is a complex sysem which can be broken down ino several areas: Prevenive fire proecion o o o Organisaional fire proecion Plan-specific fire proecion Consrucional fire proecion Defensive fire proecion This documen only deals wih he subjec of building maerial- and componen-specific properies of CLT elemens which, in reference o he fire proecion groups oulined above, are considered o belong o he consrucional fire proecion group. 1.2 The building maerial, wood when exposed o fire If he building maerial, wood is exposed o fire and hus o an elevaed supply of energy, is emperaure rises and he waer molecules embedded wihin sar o evaporae a approx. 100 C. A C, he long-chain molecules in he cell walls spli, producing gaseous and flammable compounds and he gas subsequenly eners he surface of he wood where i reacs wih oxygen in he air, and combuss. [1] These chemical compounds decompose in a process known as pyrolysis (whereby gas emissions from combusible componens in he wood burs ino flame), gradually spreading along he wood, leaving a charring area behind i. This char layer is formed from he carbonaceous residue of pyrolysis, which burns, generaing embers. This layer s properies in paricular, low densiy and high permeabiliy ac as hea insulaion and proec he underlying, undamaged wood. Page 3 of 50

5 I N T R O D U C T I O N Fig. 1: cross-secion of an 80 mm CLT elemen, originally clad wih fire proecion plaserboard, afer a large-scale fire es Figure 1 shows he cross-cu secion of a CLT elemen, clad wih fire proecion plaserboard afer a large-scale fire es. I is possible o idenify he differen layers on his cross secion: he charred area (black area), followed by he pyrolysis area (brown area) caused by he spreading fire or pyrolysis and he undamaged wood. Fig. 2: char layer of an 80 mm CLT elemen, originally clad wih fire proecion plaserboard, afer a large-scale fire es Page 4 of 50

6 R E A C T I O N - TO- F I R E P E R F O R M A N C E O F B U I L D I N G P R O D U C T S 2 Reacion-o-fire performance of building producs The reacion-o-fire performance of building producs is classified according o EN Euro classes: A1, A2, B, C, D, E, F (Crieria: igniabiliy, flame propagaion, hea release) Smoke classes: s1, s2, s3 (s1 => lowes smoke producion) Burning droples classes: d0, d1, d2 (d0 => no flaming droples) The reacion-o-fire performance of Sora Enso CLT is classified according o [26] as D-s2, d0. When using CLT for raw floors (i.e. wihou any floor srucure), D fl -s1 applies. When using flame reardans which can delay he combusion of derived imber producs and reduce he subsequen release of energy, he fire behaviour of CLT can, depending on he reardan used, be classified as class C or also B. When used oudoors wih he relaed likely effecs of humidiy and direc exposure o weaher, i mus be ensured ha he produc has he necessary properies and resisance. Page 5 of 50

7 F I R E R E S I S T A N C E O F B U I L D I N G C O M P O N E N T S 3 Fire resisance of building componens 3.1 Classificaion The performance characerisics and fire resisance duraion are defined as follows according o he classificaion sandard EN : R (Load-bearing capaciy) The performance characerisic R is assumed o be saisfied if he load-bearing funcion of a componen subjeced o a mechanical load is mainained during he required ime of fire exposure. E (Inegriy) The performance characerisic E of a separaing elemen describes is capaciy o resis exposure o fire on one side so ha he spread of fire as a resul of flames or ho gases o he side no exposed o fire is prevened. I (Insulaion) The performance characerisic I is assumed o be saisfied if he average emperaure rise over he whole of he non-exposed side caused by one-sided exposure o fire does no exceed 140 C, and he maximum emperaure a any one poin does no exceed 180 C, above ambien emperaure. Hea ransmission mus be limied so ha he non-exposed componen surface and any neighbouring maerials do no cach fire, and so ha any persons in he viciniy are proeced. Addiional characerisics are W (hermal radiaion), M (resisance o mechanical acion), C (self-closing capabiliy), S (smoke leakage), G (soo fire resisance) and K (fire proecion abiliy), which, in general, are no relevan for convenional CLT componens. The classificaion ime is graduaed in periods of 10, 15, 20, 30, 45, 60, 90, 120, 180, 240 and 360 minues. The direcion of he classified fire resisance duraion is described wih he following abbreviaions according o EN : Classificaion of façades (curain-walls) and exernal walls i o o i o i classified fire resisance duraion from inside o ouside classified fire resisance duraion from ouside o inside classified fire resisance duraion from inside o ouside and from ouside o inside Classificaion of ceilings wih independen fire resisance a b a b a b classified fire resisance duraion from op o boom classified fire resisance duraion from boom o op classified fire resisance duraion from op o boom and from boom o op Page 6 of 50

8 F I R E R E S I S T A N C E O F B U I L D I N G C O M P O N E N T S 3.2 Fire proecive cladding The designaions K 1 and K 2 for fire proecive cladding are described in accordance wih EN as follows: The erm fire proecive cladding refers o he ouermos layer on verical componens and o he lowes layer on horizonal or inclined componens. The cladding defined by he designaion K 1 or K 2 mus provide he proecion described in accordance wih EN for he layer direcly backing he fire proecive cladding hroughou he corresponding classificaion period (K 1 : 10 min.; K 2 : 10, 30 or 60 min.). For example, fire proecive cladding wihou an underlying caviy wih he classificaion K 2 60 mus provide he following proecion for a period of 60 minues: During he classificaion period, he fire proecive cladding mus no collapse in whole or in par. The average emperaure recorded on he underside of he supporing plae (on which he fire proecive cladding o be classified is esed) mus no exceed he ambien emperaure by more han 250 C. The maximum emperaure recorded on (a any one poin) on he underside of he supporing plae (on which he fire proecive cladding o be classified is esed) mus no exceed he ambien emperaure by more han 270 C. Afer he es, no burned or charred maerials should be apparen on any one poin of he supporing plae (on which he fire proecive cladding o be classified is esed). Informaion on he fire proecive cladding o be classified can be obained from gypsum plaserboard manufacurers, among ohers. 3.3 Fire resisance of CLT componens Componens wih high fire resisance can be produced wih muliple layer CLT elemens. For example, wih a non-clad, hree-layer CLT elemen, he fire resisance REI 60 is already obained, and wih a CLT elemen clad wih a single layer of plaserboard, he fire resisance REI 90 is obained. In principle, increased requiremens for fire resisance can be compensaed by he following measures: Increase he hickness of he CLT elemen Increase he number of layers of he CLT elemen Apply he corresponding cladding The verificaion of fire resisance of imber componens can eiher be based on classificaion repors in accordance wih EN on he basis of large-scale fire ess, or on calculaions according o EN , performed in conjuncion wih he respecive naional applicaion documens. Page 7 of 50

9 V E R I F I C A T I O N O F F I R E R E S I S T A N C E 4 Verificaion of he fire resisance of CLT elemens based on classificaion repors according o EN Sora Enso commissioned differen accredied es insiues o es he fire resisance of differen CLT elemens wih differen componen designs according o EN or EN The resuls of he classificaion repors ([27], [28], [29], [30], [31], [32] and [36]) from he fire resisance ess, performed in accordance wih EN , are as follows. 4.1 CLT exernal wall srucures Classificaions of he fire resisance REI 90 of load-bearing cross-laminaed imber elemens as exernal wall elemens: Inernal cladding 12.5 mm fire proecion plaserboard 12.5 mm fire proecion plaserboard 12.5 mm fire proecion plaserboard 12.5 mm fire proecion plaserboard 12.5 mm fire proecion plaserboard 12.5 mm fire proecion plaserboard Service caviy Cross-laminaed imber elemen Exernal cladding Designaion Lamella srucure CLT 100 C3s mm wood wool slab, 15 mm plaser CLT 100 C3s mm rock fibre, 4 mm plaser CLT 100 C5s mm wood wool slab, 15 mm plaser CLT 100 C5s mm rock fibre, 4 mm plaser rock fibre (40 mm) rock fibre (40 mm) CLT 100 C3s mm wood wool slab, 15 mm plaser CLT 100 C3s mm rock fibre, 4 mm plaser Table 1: classificaions of he esed componens in accordance wih [32] Tes load [kn/m] Classificaion i o 35 REI REI REI REI REI REI 90 Page 8 of 50

10 V E R I F I C A T I O N O F F I R E R E S I S T A N C E 4.2 CLT wall srucures Classificaions of he fire resisance REI 60 of load-bearing cross-laminaed imber elemens as wall elemens: Cladding Service caviy Cross-laminaed imber elemen Tes load Classificaion Designaion Lamella srucure [kn/m] CLT 100 C3s REI 60 CLT 100 C5s REI 60 Table 2: classificaions of he esed componens in accordance wih [27] Classificaions of he fire resisance REI 90 of load-bearing cross-laminaed imber elemens as wall elemens: Cladding Caviy Cross-laminaed imber elemen Tes load Classificaion Designaion Lamella srucure [kn/m] 12.5 mm fire CLT 100 C3s REI 90 proecion plaserboard 12.5 mm fire CLT 100 C5s REI 90 proecion plaserboard 12.5 mm fire proecion plaserboard rock fibre (40 mm) CLT 100 C3s REI 90 Table 3: classificaions of he esed componens in accordance wih [28] Cladding Caviy Cross-laminaed imber elemen Tes load Classificaion Designaion Lamella srucure [kn/m] 35 mm ProCrea clay board, 5 mm ProCrea clay plaser wih reinforcemen fabric, 5 mm ProCrea clay plaser CLT 140 C5s REI 90 Table 4: classificaion of he esed componens in accordance wih [36] Classificaions of he fire resisance REI 120 of load-bearing cross-laminaed imber elemens as wall elemens: Cladding Caviy Cross-laminaed imber elemen Tes load Classificaion Designaion Lamella srucure [kn/m] 12.5 mm fire proecion plaserboard rock fibre (40 mm) CLT 100 C3s REI 120 Table 5: classificaion of he esed componens in accordance wih [29] Page 9 of 50

11

12 V E R I F I C A T I O N O F F I R E R E S I S T A N C E 5 Verificaion of he fire resisance of CLT elemens based on calculaions according o EN :2011 (Eurocode 5) For he required ime of fire exposure, i mus be demonsraed ha: E d, fi Rd,, fi where: E d,fi R d,,fi is he design value of acions for he fire siuaion (= load effec) (Wih maerials oher han wood, hermal expansion mus also be aken ino accoun.) is he corresponding design resisance in he fire siuaion (= resisance) 5.1 Verificaion mehod for acions in he fire siuaion according o EN :2011 The design value for acions in he fire siuaion should be deermined for ime = 0 using combinaion facors 1,1 or 2,1 according o EN :2002, clause (See also EN , clause ) E da Gk, j" " P" " Ad " " Qk,1 ( 1,1 oder 2,1)" " 2, i Qk, i where: G k,j is he characerisic value of a permanen acion j P is he decisive represenaive value of a pre-load A d is he design value of an excepional acion Q k,1 is he characerisic value of a decisive variable acion 1 Q k,i is he characerisic value of a decisive variable acion i 1 is he combinaion facor for frequen values of variable acions is he combinaion facor for quasi-permanen values of variable acions 2 I is up o he user o choose/use 1,2 or 2,1. Page 11 of 50

13 V E R I F I C A T I O N O F F I R E R E S I S T A N C E For simpliciy, he design value for acions in he fire siuaion E d,fi from he calculaion of he design value for acions a normal emperaure E d may be deermined hus: E d, fi E fi d where: E d,fi fi E d is he design value for acions for he fire siuaion is he reducion facor for he design value of acions in he fire siuaion is he design value for acions a normal emperaure for he fundamenal combinaion of acions For he load combinaion in accordance wih EN , he reducion facor fi should be aken as follows, whereby he smalles value is given by he following wo equaions: Gk fi Qk,1 fi G Q G k Q,1 k,1 Gk fi Qk,1 fi G Q G k Q,1 k,1 where: Q k,1 G k G Q,1 fi is he characerisic value of he leading variable acion is he characerisic value of permanen acions is he parial safey facor for permanen acions is he parial safey facor for he leading variable acion is he combinaion facor for frequen values of variable acions in he fire siuaion, given eiher by 1,1 or 2,1, see EN is a reducion facor for unfavourable permanen acions G (see EN , clause A.1.3.1) As a simplificaion, for he reducion facor fi, as an alernaive o he above equaion, he recommended value is fi = 0.6 according o EN :2011, clause Excepions here are areas wih larger imposed loads according o caegory E given in EN :2002, where he recommended value is fi = 0.7. When comparing he opions for deermining acions, i is clear ha he simplified assumpion wih he acion E d,fi resuls in a greaer load han he acions in he excepional design siuaion. Page 12 of 50

14 V E R I F I C A T I O N O F F I R E R E S I S T A N C E 5.2 Verificaion mehod for mechanical resisance in he fire siuaion according o EN :2011 For verificaion of mechanical resisance, he design values of srengh and siffness properies shall be deermined from: f d k f20, fi mod, fi M, fi where: f d,fi k mod,fi f 20 f k k fi M,fi is he design value of srengh in fire is he modificaion facor in he fire siuaion for he reduced cross-secion mehod: k mod,fi = 1.0 (as per EN ) is he 20% fracile value of a srengh propery a normal emperaure; f 20 = k fi f k is he 5% fracile value of a srengh propery is he coefficien for convering 5% o 20% fracile values; k fi for CLT = 1.15 (as per EN ) is he parial safey facor for imber in fire M,fi = 1.0 (as per EN ) For he calculaion in he fire siuaion, insead of he 5% fracile values, he 20% fraciles are used. The reason for his assumpion lies in he exremely low probabiliy of occurrence of a fully developed fire during he lifeime of a supporing srucure, and does no depend on he maerial. [25] (Hence he coefficien for convering he fracile value k fi wih 1.15.) S d k S20, fi mod, fi M, fi where: S d,fi k mod,fi S 20 S 05 k fi M,fi is he design value of he siffness propery (modulus of elasiciy or shear modulus) in he fire siuaion is he modificaion facor in he fire siuaion for he reduced cross-secion mehod: k mod,fi = 1.0 (as per EN ) is he 20% fracile of a siffness propery (modulus of elasiciy or shear modulus) a normal emperaure S 20 = k fi S 05 is he 5% fracile of a siffness propery (modulus of elasiciy or shear modulus) a normal emperaure is he coefficien for convering 5% o 20% fracile values; k fi for CLT = 1.15 (as per EN ) is he parial safey facor for imber in fire M,fi = 1.0 (as per EN ) Page 13 of 50

15 V E R I F I C A T I O N O F F I R E R E S I S T A N C E 5.3 Charring raes for Sora Enso CLT During exposure o fire and o he resuling effec of emperaure on he CLT cross-secion, he use of polyurehane adhesives beween individual layers can lead o sofening. A possible consequence of his may be ha small secions of he hea-insulaing char layer fall off, and he proecive funcion of his layer may be los a cerain poins. [2] Therefore, in he case of ceiling elemens and oher horizonal componens, possible delaminaions mus be aken ino accoun, and, for he subsequen fire-exposed layers, i is necessary o mahemaically esimae an increased charring rae unil he formaion of a new 25 mm-hick char layer Design value of charring raes 0 for CLT on surfaces which are unproeced hroughou he duraion of he fire The following charring raes for Sora Enso CLT were deermined as par of [8] by he accredied insiue Holzforschung Ausria, and may be used for he calculaion of he fire resisance of consrucions wih differen loads and/or layer hicknesses according o EN (wih reference o he respecive naional annex). Ceiling and roof elemens (horizonal componens): o 0.65 mm/min., if only one layer is affeced by exposure o fire. [33] o 1.3 mm/min. for any addiional layers affeced by exposure o fire unil charring or he formaion of a 25 mm-hick char layer. Thereafer, a charring rae of 0.65 mm/min. can be applied up o he nex bonded join. [33] Fig. 3: diagram illusraing an example of charring or he charring rae of a horizonal CLT componen (CLT 180 L5s), which explains he mahemaically esimaed charring rae of 1.3 mm/min. for each addiional layer affeced by fire unil he formaion of a new 25 mm-hick char layer. Page 14 of 50

16 V E R I F I C A T I O N O F F I R E R E S I S T A N C E Wall elemen (verical componens): o 0.63 mm/min., if only one layer is affeced by exposure o fire. [33] o 0.86 mm/min. for each addiional layer affeced by exposure o fire. [33] Fig. 4: diagram showing an example of charring or he charring rae of a verical CLT componen (CLT 100 L5s), which explains he mahemaically esimaed increased charring rae of 0.86 mm/min. from he second layer affeced by fire. Page 15 of 50

17 V E R I F I C A T I O N O F F I R E R E S I S T A N C E Design value of charring raes 0 for CLT on surfaces which are iniially proeced from exposure o fire by gypsum plaserboard The fire resisance raing of componens is deermined during exposure o fire on he inside of a room predominanly by inerior cladding. To increase he fire resisance of srucures such as wall, ceiling or roof elemens, plaser building maerials/gypsum plaserboards are generally used as, even if hey are no very hick, hey provide effecive proecion. Effecive proecion is based paricularly on he combined crysal waer in he panels gypsum core which has a concenraion of approx. 20%. Energy is consumed by he evaporaion of his crysal waer, and a proecive seam curain is also formed on he fire-exposed side of he componen. In addiion o delaying he spread of fire, he dehydraed gypsum layer also acs as insulaion hrough he declining hermal conduciviy. Fire proecion plaserboard also conains glass fibre which reinforces he gypsum core and ensures srucural cohesion when exposed o fire. [3] Fig. 5: wo-ply fire proecion plaserboard cladding exposed o fire during a large-scale fire es Figure 5 illusraes he behaviour of fire proecion plaserboard when exposed o fire; in his case here are wo layers of cladding. As can be seen, afer crazing and deaching of he char layer, as ime progresses, large gaps appear beween he joins, he join plaser compound fails and he firs secion of he firs plaserboard layer falls off. If larger panel secions fall away from he firs layer, crazing also occurs in he second layer. Afer gaps appear in his layer s joins, he flames spread hrough he increasing gaps in he joins owards he underlying CLT which leads o he producion and emission of wood gas. Charring sars on he iniially proeced CLT elemen. The following correlaion or equivalence wih regard o gypsum plaserboard designaions should be noed: Designaion according o EN 520 Gypsum plaserboard, ype A Gypsum plaserboard, ype F or DF Gypsum plaserboard, ype H2 Gypsum plaserboard, ype DFH2 Designaion according o ÖNORM B 3410 and DIN Plaserboard cladding Fire proecion plaserboard Plaserboard cladding waerproofed Fire proecion plaserboard waerproofed Table 8: comparison of gypsum plaserboard designaions by EN 520 and ÖNORM B 3410 or DIN Page 16 of 50

18 V E R I F I C A T I O N O F F I R E R E S I S T A N C E In he case of iniially proeced componens, he ime of sar of charring behind he proecive layer or cladding ch and he failure ime of he proecive cladding f is essenial. According o EN :2011, he following mus be aken ino accoun: The sar of charring is delayed unil ime ch ; Charring can occur before failure of he fire proecive cladding, however unil he failure ime f, he charring rae is lower han he value according o [22], able 3.1 or he value according o [33]; The charring rae afer he failure ime f of he fire proecive cladding unil ime a is greaer han he value according o [22], able 3.1 or he value according o [33]; The charring rae from ime a, where he charring deph corresponds o he lowes value eiher he charring deph of a similar componen wihou fire proecive cladding or 25 mm again akes he values according o [22], able 3.1 or he values according o [33]. The following illusraions from EN :2011 are provided o aid undersanding of he above poins: Key: 1 Relaionship for componens which are unproeced hroughou he ime of fire exposure wih he noional charring rae n (or 0) 2 Relaionship for iniially proeced componens afer failure of he fire proecive cladding 2a Afer he fire proecive cladding has fallen off, charring sars a an increased rae 2b Afer he charring deph exceeds 25 mm or he ime a is exceeded, he charring rae reduces o he normal rae Fig. 6: illusraion of he charring deph depending on he ime for ch = f and a charring deph of 25 mm a ime a [22] Key: Fig. 7: illusraion of he charring deph depending on he ime for ch < f [22] 1 Relaionship for componens which are unproeced hroughou he ime of fire exposure wih he noional charring rae n (or 0) 2 Relaionship for iniially proeced componens on which charring sars before failure of he fire proecive cladding 2a Charring sars a ch, a a reduced rae for as long as he fire proecive cladding remains inac 2b Afer he fire proecive cladding falls off, charring sars a an increased rae 2c Afer he charring deph exceeds 25 mm or he ime a is exceeded, he charring rae reduces o he normal value Page 17 of 50

19 V E R I F I C A T I O N O F F I R E R E S I S T A N C E Charring raes for iniially proeced componens For ime ch f, according o [22], he charring raes given in EN , able 3.1 or according o he saemen of exper opinion of he Holzforschung Ausria should be muliplied by a facor k 2 ; for single layer gypsum plaserboard, ype F, his is calculaed as: k 2 1 0, 018 h p where: h p is he hickness of he layer in mm For several layers of gypsum plaserboard, ype F, h p should be aken as he hickness of he inner layer. If he imber componen is proeced by rock fibre bas (hickness: 20 mm, bulk densiy: 26 kg/m 3, meling poin: 1000 C), he facor k 2 may be aken from able 9. For hicknesses beween 20 and 45 mm, linear inerpolaion may be applied. Table 9: values of k 2 for imber componens proeced by rock fibre bas [22] For he sage afer failure of he fire proecive cladding given by f a, according o [22], he charring raes given in EN , able 3.1 or according o he saemen of exper opinion of Holzforschung Ausria should be muliplied by a facor k 3 = 2. For a, he charring raes should be applied wihou muliplicaion by he facor k 3. The ime limi a (see figure 6) should for ch = f, in accordance wih [22], be aken as: a 2 f min 25 k3 n f Or for ch < f : a 25 ( f ch ) k2 n k 3 n f where: n is he design value of he noional charring rae in mm/min. (In he case of one-dimensional charring, n is replaced by 0.) Page 18 of 50

20 V E R I F I C A T I O N O F F I R E R E S I S T A N C E Sar of charring on iniially proeced componens Single layer gypsum plaserboard, ype A, F or H: For claddings consising of one layer of gypsum plaserboard, ype A, F or H, according o EN 520, ouside of joins or a locaions adjacen o filled joins, or unfilled gaps wih a widh of 2 mm or less, in accordance wih [22], he sar of charring ch should be aken as: ch 2,8 h 14 p In locaions adjacen o joins wih unfilled gaps wih a widh of more han 2 mm, he ime of sar of charring should be calculaed as: ch 2,8 h 23 where: p ch h p is he ime of sar of charring of a proeced componen in minues is he hickness of he fire proecive cladding in mm Two-layer gypsum plaserboard, ype A or H: For claddings consising of wo layers of gypsum plaserboard, ype A or H in accordance wih EN 520, according o [22], he ime of sar of charring ch should be deermined according o he formula in , where he hickness h p is aken as he hickness of he ouer layer and 50% of he hickness of he inner layer. This is subjec o he condiion ha he spacing of faseners in he inner layer is no greaer han he spacing of faseners in he ouer layer. Claddings consising of wo layers of gypsum plaserboard, ype F: For claddings consising of wo layers of gypsum plaserboard, ype F in accordance wih EN 520, according o [22], he ime of sar of charring ch should be deermined according o he formula in , where he hickness h p is aken as he hickness of he ouer layer and 80% of he hickness of he inner layer. This is subjec o he condiion ha he spacing of faseners in he inner layer is no greaer han he spacing of faseners in he ouer layer. Caviy insulaion maerial: If he imber componen is proeced by rock fibre bas (hickness: 20 mm, bulk densiy: 26 kg/m 3, meling poin: 1000 C), for he ime of sar of charring ch, he following equaion mus also be aken ino accoun: 0,07( 20) ch h ins ins where: ch h ins is he ime unil he sar of charring of a proeced componen in minues is he insulaion maerial hickness in mm ins is he insulaion maerial bulk densiy in kg/m 3 Page 19 of 50

21 V E R I F I C A T I O N O F F I R E R E S I S T A N C E Failure ime of fire proecive cladding In principle, he charring or mechanical degradaion of he cladding maerial, he spacing of, and disances beween, faseners and/or a possibly insufficien peneraion lengh of faseners ino he uncharred cross-secion could be responsible for he failure of he fire proecive cladding. Cladding consising of gypsum plaserboard, ype A or H: For gypsum plaserboard, ype A or H in accordance wih EN 520, according o [22], he failure ime f is equal o he ime a he sar of charring ch. For gypsum plaserboard, ype A or H, afer he sar of charring and afer he cladding simulaneously falls off, charring occurs a double he rae unil ime a. Afer formaion of a 25 mm-hick char layer, he charring rae reduces o he normal rae. (See also fig. 6) f ch Cladding consising of gypsum plaserboard, ype F: However, in he case of gypsum plaserboard, ype F or fire proecion plaserboard, according o [22], here is less charring from he sar of charring ch o he ime f. Unil he subsequen formaion of a 25 mm-hick char layer, charring occurs a double he rae, afer which, he charring rae reduces o he normal rae. (See also Fig. 7.) EN :2011 does no provide any informaion regarding he failure ime of gypsum plaserboard, ype F or fire proecion plaserboard. According o ÖNORM B :2011 (Ausrian naional specificaions), he failure imes f for cladding consising of fire proecion plaserboard in accordance wih ÖNORM B 3410 or gypsum plaserboards, ype DF according o EN 520 and gypsum fibreboard GF-C1-W2 according o EN can be deermined as follows: Wall componens: f 2,2 h 4 Ceiling componens: 1,4 h 6 where: f p p f h p is he failure ime of he fire proecive cladding in minues is he hickness of he fire proecive cladding in mm In deermining he failure ime of muliple-layer cladding consising of gypsum plaserboard, ype F, he rules specified in secion apply correspondingly, according o which, he hickness h p corresponds o he hickness of he ouer layer and o 80% of he hickness of he inner layer. Page 20 of 50

22 V E R I F I C A T I O N O F F I R E R E S I S T A N C E Peneraion lengh of faseners for gypsum plaserboard In addiion o hermal degradaion of he cladding maerial, he fire proecive cladding can also fall off due o he pull-ou failure of faseners. According o [22], he required minimum lengh of he faseners should also be deermined in order o eliminae he fac ha pull-ou of he faseners is a relevan facor for he failure of he fire proecive cladding. The minimum peneraion lengh of he fasener l a ino he unburn cross-secion should be aken as 10 mm. The required peneraion lengh of he fasener l f, req is calculaed as follows: l f, req hp dchar, 0 l a where: h p d char,0 l a is he panel hickness in mm is he charring deph in he imber componen is he minimum peneraion lengh of he fasener ino he unburn wood For more informaion on cladding faseners/peneraion lenghs, see [22], secion Failure imes f of fire proecion plaserboard on Sora Enso CLT confirmed by saemens of exper opinion: In addiion o he equaions demonsraed above and specified in [22] and [25] o deermine he sar of charring ch and he failure ime f of gypsum plaserboards, Sora Enso has a saemen of exper opinion on failure imes which mus be referred o during dimensioning according o EN According o [35], based on various ess, he failure imes f lised in able 10 were given for fire proecion plaserboards in accordance wih ÖNORM B 3410 or gypsum plaserboards, ype DF in accordance wih EN 520. (Compare he calculaed values according o he equaions of [25], which were originally worked ou for imber frame srucures.) Fire proecion plaserboard CLT wall srucure (verical componen) HFA saemen ON B of exper opinion f [min.] f [min.] Table 10: failure imes for fire proecion plaserboards or gypsum plaserboards, ype DF direcly applied o CLT elemens in accordance wih [35] (cf. he failure imes calculaed according o EN ) The failure imes given in able 10 only apply o fire proecion plaserboards or gypsum plaserboards, ype DF direcly applied o Sora Enso CLT elemens. The fire proecion plaserboard mus be applied and sealed according o he manufacurer s insrucions. [35] Page 21 of 50

23

24 V E R I F I C A T I O N O F F I R E R E S I S T A N C E Noe *2: The value specified for d 0 = 7 mm is based on [26] (relaing o [22]). The value d 0 of 7 mm (for he simplified calculaion mehod of he reduced cross-secion mehod) is currenly being discussed by scieniss around he world, however no unified opinion has been esablished. Possible naional regulaions on d 0 mus be aken ino accoun. Wih regard o assumpions abou charring raes for Sora Enso CLT, he following mus be observed: When using CLT for fla componens (wall and ceiling srucures), one-dimensional charring raes in accordance wih [33] (see secion 5.3.1) should be used. When using CLT for suppors (edgewise), for example, proceed according o [22], secion In doing so, for CLT cross-secions wih original widhs which do no mee requiremens, increased charring raes should be expeced. When verifying he load-bearing capaciy in he fire siuaion of Sora Enso CLT componens, he following mus be observed: Charring on boh sides mus be aken ino accoun on load-bearing elemens wih no separaing funcion. [22] Possible addiional, eccenric load applicaions due o one-sided charring mus be aken ino accoun paricularly on hinner CLT elemens. Residual cross-secions of layers 3 mm are no used in he remaining calculaions. (This assumpion akes ino accoun he generally non-linear naure of he char line.) The remaining calculaion seps and verificaions are performed in he same way as he cold calculaions. Page 23 of 50

25 V E R I F I C A T I O N O F F I R E R E S I S T A N C E 5.5 Deermining he inegriy (E) and insulaion (I) of CLT elemens The following opions exis for verificaion of inegriy (E) and insulaion (I): Calculaion mehod according o EN :2011, annex E ([22]) Model according o ÖNORM B :2011, 14.3 ([25]) or he European guideline Fire safey in imber buildings ([9]) or he disseraion by Ms Schleifer ([4]) Srucures wihou addiional verificaions according o ÖNORM B :2011 ([25]) Verificaion of he inegriy and insulaion of CLT can be performed using he model specified in ÖNORM B :2011 ([25]) or in he European echnical guideline Fire safey in imber buildings ([9]) and elaboraed by [4], which have he same approach/suppor he same heory. If we compare his model wih he calculaion mehod specified in EN :2011, annex E, he possibiliy of an unlimied variaion of maerials and number of layers can be considered o be a significan advanage. Exended mehod for deermining he inegriy (EI) of wall and ceiling srucures in accordance wih ÖNORM B :2011 ([25]) or he European guideline Fire safey in imber buildings ([9]) In principle, he following applies o he calculaion: The influence of emperaure according o he uniform emperaure curve as per EN provides he basis for he calculaion model. According o [25], he calculaion model is limied o a fire resisance duraion of 60 minues. Validaion calculaions performed by he accredied insiue Holzforschung Ausria as par of large-scale fire ess show ha his model can also be used for a fire resisance duraion of 90 minues. [5] The requiremen for inegriy (E) is considered saisfied if he requiremen for he insulaion (I) crierion is shown o be posiive. The requiremen for insulaion (I) is saisfied if he average emperaure rise on he unexposed side of he componen does no exceed 140 C, or 180 C a any one poin (above he ambien emperaure). In order o ensure he calculaed fire resisance or inegriy, in he case of composie imber componens, he surplus insulaion mus be prevened from falling ou afer failure of he cladding by mechanical means, where appropriae. I mus also be ensured, hrough correc insallaion according o he manufacurer s insrucions (e.g. informaion relaing o spacing beween faseners and peneraion lenghs), ha he cladding on he unexposed side canno fall off a an early sage. Page 24 of 50

26 V E R I F I C A T I O N O F F I R E R E S I S T A N C E The componen may be composed of any of he following panel and insulaion maerials and may be designed wih a caviy: Panel maerials (faseners according o he manufacurer s insrucions): Solid wood panels of a leas C 24 in accordance wih EN 338 OSB panels in accordance wih EN 300 Paricle board (chipboard) in accordance wih EN 309 Gypsum plaserboard, ype A, H and F in accordance wih EN 520 Gypsum fibreboard in accordance wih EN Insulaion maerial (surplus insallaion according o he manufacurer s insrucions): Rock fibre in accordance wih EN Glass wool in accordance wih EN The required inegriy of a componen is considered saisfied if he following equaion is saisfied: ins req where: ins req is he ime unil failure of he separaing funcion of he enire componen, in minues. is he required fire resisance duraion for he separaing funcion of he enire componen, in minues The insulaing and proecive layers of a componen Based on simulaion calculaions or FE modelling, i was demonsraed ha he insulaion (I) requiremens were saisfied. If we consider a imber srucure wih muliple layers, he individual layers are arranged wih a proecive (proecive in erms of he underlying layers) and insulaing funcion (las layer of he unexposed side). [4] Fig. 8: design of a muliple layer imber consrucion o define he proecive and insulaing layers [4] Page 25 of 50

27 V E R I F I C A T I O N O F F I R E R E S I S T A N C E The ime of he separaing funcion (EI) of he componen under consideraion is he ime unil he emperaure crierion T MW / T Max = 140 / 180 C is reached on he unexposed side. This crierion wih he maximum emperaure o be mainained of T = 160 C (20 C room emperaure K) is only relevan for he unexposed side of he las layer. The preceding proecive layers mus saisfy he cladding crierion in accordance wih EN , whereby he emperaure crierion T MW / T Max = 250 / 270 C mus be saisfied. Thus, in applying he average value, a maximum emperaure o be mainained of T = 270 C (20 C room emperaure K) is obained. When he emperaure crierion of 270 C is reached (= proecion ime pro,i is reached), i is assumed ha he esed layer i will fall off he srucure and he proecion ime of he direcly underlying layer pro,i+1 begins. Fig. 9: mehod for deermining he conribuions of individual layers [4] As a resul, he proecive layers lose heir proecive funcion as soon as a emperaure of T = 270 C is reached on heir unexposed side. The ime of he separaing funcion of a complee componen ins is hus obained by adding ogeher he conribuions of he individual layers he proecion imes of he proecive layers and he insulaion ime of he insulaing layer while observing he above-menioned emperaure crieria of 270 or 160 C. ins, pro, i1 ins i where: ins pro,i ins,i is he ime unil failure of he separaing funcion of he enire componen, in minues is he proecion ime of he layer i, in minues is he insulaion ime of he layer i, in minues Page 26 of 50

28 V E R I F I C A T I O N O F F I R E R E S I S T A N C E When deermining he proecion and insulaion ime, i is imporan o noe ha he preceding and backing layers influence he layer under invesigaion depending on is posiion in he componen. This is aken ino accoun in he calculaion model wih he posiion coefficien k pos. In he process, k pos,exp is he posiion coefficien resuling from he influences of he layers preceding he layer under invesigaion and k pos,unexp is he posiion coefficien resuling from he influences of he layers backing he layer under invesigaion. The insulaion and proecion ime of individual layers is deermined wih he following equaions: pro, i ( pro,0, i kpos,exp, i kposun, exp, i i ) kj, i where: pro,i pro,0,i k pos,exp,i k pos,unexp,i,i k j,i is he proecion ime of he layer under invesigaion i, in minues is he basic insulaion ime of he layer i, in minues is he posiion coefficien for he layer under invesigaion i (influences from he preceding layer) is he posiion coefficien for he layer under invesigaion i (influences from he backing layer) is he ime difference for he layer under invesigaion i, in minues. (To ake ino accoun he influence of he preceding gypsum plaserboard, ype F or gypsum fibreboard) is he join coefficien for he layer under invesigaion i ins, i ( ins,0, i kpos,exp, i i ) kj, i where: ins,i ins,0,i k pos,exp,i,i k j,i is he insulaion ime of he layer under invesigaion i, in minues. is he basic insulaion ime of he layer i, in minues. is he posiion coefficien for he layer under invesigaion i (influences from he preceding layer) is he ime difference (= delayed fall off ime) for he layer under invesigaion i, in minues; (To ake ino accoun he influence of he preceding gypsum plaserboard, ype F or gypsum fibreboard) is he join coefficien for he layer under invesigaion i Page 27 of 50

29

30 V E R I F I C A T I O N O F F I R E R E S I S T A N C E Calculaing he posiion coefficien k pos To enable any combinaion of layers in a imber srucure, he influences of adjacen layers on he layer under invesigaion mus be aken ino accoun. The influence of he preceding layer is described or calculaed wih posiion coefficien k pos,exp and he influence of he underlying layer wih posiion coefficien k pos,unexp Posiion coefficien k pos,exp for aking ino accoun he influence of he preceding layers The posiion coefficien k pos,exp is assumed o be 1.0, if he layer under invesigaion i is exposed o fire from he sar of he fire and if i is no proeced by any preceding layers. If he layer under invesigaion i is proeced from direc exposure o fire by preceding layers, he posiion coefficien k pos,exp mus be defined according o he equaions in he following able. When deermining he posiion coefficien k pos,exp, he sum of he proecion imes of he preceding layers pro,i-1, he maerial of he layer under invesigaion i, he hickness of he layer under invesigaion i and he bulk densiy of he layer under invesigaion i mus be aken ino accoun. However, he properies of he layer under invesigaion i have already been calculaed during deerminaion of he basic proecion ime pro,0,i or he basic insulaion ime ins,0,i and herefore he posiion coefficien k pos,exp for cladding and insulaion can be calculaed depending on he sum of he proecion imes of he preceding layers pro,i-1 and he basic ime (basic proecion ime pro,0,i or basic insulaion ime ins,0,i ). Maerial k pos,exp for ins,i k pos,exp for pro,i Rock fibre see formulae for glass wool see formulae for cladding Glass wool for h i 40 mm for for Cladding for Table 12: equaions o deermine he posiion coefficien k pos,exp [25] for When using he equaions in able 12, i mus be considered wheher an insulaion ime or a proecion ime should be calculaed, i.e. he basic proecion ime or insulaion ime mus be used accordingly for he basic ime. Page 29 of 50

31

32 V E R I F I C A T I O N O F F I R E R E S I S T A N C E Maerial of layer under invesigaion Gypsum plaserboard in accordance wih ÖNORM EN 520 (applies o: ype A, ype H, ype F, ype DF) k pos,unexp wih underlying cladding k pos,unexp wih underlying insulaion Gypsum fibreboard in accordance wih ÖNORM EN (applies o: GF-C1-W2) Solid wood board in accordance wih ÖNORM EN Paricle board (chipboard) in accordance wih ÖNORM EN 312 OSB board in accordance wih ÖNORM EN 300 Rock fibre in accordance wih ÖNORM EN Glass wool in accordance wih ÖNORM EN Table 14: equaions o deermine he posiion coefficien k pos,unexp [25] 1,0 Regarding he posiion coefficiens k pos,exp and k pos,unexp, caviies mus also be aken ino accoun in srucures if hey are a leas 40 mm hick: For layers preceded by caviies, he posiion coefficien k pos,exp mus firs be muliplied by he facor 1.6. o If he caviy is proeced from direc exposure o fire by gypsum plaserboard, ype F or gypsum fibreboard, for he layer on he unexposed side of he caviy, he ime difference,i according o he equaions in able 13, line 2, mus also be muliplied by he facor 3. The posiion coefficien k pos,unexp for cladding wih an underlying caviy is deermined based on he equaions in able 14, column 3. The posiion coefficien k pos,unexp for insulaion wih an underlying caviy is assumed o be 1.0. Page 31 of 50

33 V E R I F I C A T I O N O F F I R E R E S I S T A N C E Deermining he join coefficien k j,i According o [22], joins wih a gap widh greaer han 2 mm are no permied and may no be used in he calculaion model. In addiion, unfilled joins on imber cladding and unsealed joins on gypsum plaserboard are no permied on he cladding no exposed o fire. For simpliciy, i is assumed ha, apar from cladding on he side no exposed o fire, he join coefficien for imber panels and insulaion wih join deailing shown in he able below, and for bu-joined sealed joins on gypsum plaserboards is 1.0. [4] Therefore, in he calculaion model for verificaion of inegriy, only join coefficiens for he layer on he side of he componen which is no exposed o fire are calculaed, which can be referred o in able 15. Maerial Join ype k j,i for ins,i k j,i for pro,i Timber maerials in accordance wih ÖNORM EN Solid wood in accordance wih ÖNORM EN 338 Glass wool or rock fibre in accordance wih ÖNORM EN ,3 1,0 0,4 1,0 0,6 1,0 Gypsum plaserboard in accordance wih ÖNORM EN 520 (applies o: ype A, ype H, ype F, ype DF) Gypsum fibreboard in accordance wih ÖNORM EN (applies o: GF-C1-W2) all maerials filled and join ypes in accordance wih he manufacurer's insrucions (filled) Join ypes oher han he above ypes Cuous Table 15: join coefficien k j,i for non-fire-exposed cladding [25] 0,8 1, Calculaion model/applicaion for CLT Due o heir layered srucure, muliple layer solid wood panels canno be compared o single-layer solid wood panels. Therefore, when deermining he inegriy of CLT elemens using his model, each individual layer mus be regarded as an independen panel or layer. [4] The inegriy of cross-laminaed imber elemens can hus be deermined as he sum of he proecion imes and he insulaion ime of he individual layers using he specified basic proecion imes, basic insulaion imes and posiion coefficiens for single layer solid wood panels. Excluding he sligh fall off of charred layers, CLT elemens can be regarded as single-layer solid wood panels. The join coefficien in he area of elemen can always be calculaed wih k j,i = 1.0. However, when verifying he elemen join area, depending on he join ype of he enire elemen, he join coefficien k j,i from able 15 should be used for all layers. [6] Page 32 of 50

34

35 T E C H N I C A L F I R E P R O T E C T I O N O F D E T A I L I N G Sora Enso CLT componen connecions Componen connecions (e.g. wall-o-ceiling, wall corner joins and wall T-joins) have he same fire resisance as individual componens as long as he connecions and joins mee he requiremens of he corresponding sandards (ÖNORM B 2330 in Ausria). o Load-ransferring connecions o Accuraely fiing fire proecive cladding [34] In he case of connecions of ceiling-o-wall componens, i mus be ensured ha he srucural funcion of he suppor mainains he required fire resisance duraion. 6.2 Insallaions and fixures Fiing wiring insallaions In elemens forming a fire secion, wiring insallaions mus be fied in an insulaed facing/service caviy. Direc insallaion in he CLT elemen is only permied if a special verificaion is performed. However, in he case of elemens which do no form a fire secion, provided ha he dimensions/designs are no greaer han hree sockes or a disribuion box, wiring can be insalled in channels cu direcly ino he CLT elemen. The hickness of he residual cross-laminaed imber elemens mus no be reduced by more han half in his localised area. Sockes posiioned on he opposie side mus be saggered by 20 cm. Alernaively, professionally insalled and esed fire proecion sockes may also be used. [34] Holes for assembly If cu-ous/holes are made in CLT elemens o aach various lifing devices, hey mus be sealed wih wooden plugs or filled wih rock fibre (meling poin 1000 C) and do no affec he classified fire resisance of he CLT elemen. [34] Insallaion of windows and doors If fire resisance requiremens for windows and/or doors are laid down, esed sysems mus be used in compliance wih he manufacurer s insrucions. In he case of requiremens for cladding crierion, he reveals mus be performed accordingly. [34] 6.3 Fire-resisan sealing in CLT consrucion Someimes, he consrucion of domesic insallaions also involves componens which form a fire secion (wall or ceiling srucures). The inersecions required for his mus be sealed in erms of fire proecion so ha he required fire resisance of he inersecing componen is no affeced. This means ha he fire proecion seal of an inersecion mus comply wih he fire resisance raing of he componen o be inerseced. Page 34 of 50

36 T E C H N I C A L F I R E P R O T E C T I O N O F D E T A I L I N G The planning and implemenaion of seals usually involves several subsecions. Corresponding advanced planning is essenial and indispensable for qualiaive separaion measures. For more informaion on he subjec of Fire-resisan sealing in imber srucures, Sora Enso refers he reader o [7]. Large-scale fire ess required for [7] for wall and ceiling componens (respecively 90 minues es duraion) were performed using Sora Enso CLT elemens. Deailed soluions for he insallaion, fasening and connecion of various sealing sysems for imber consrucions have been developed on he basis of hese es resuls and are explained in [7]. 6.4 Faseners To achieve a good fire proecion join, he seel componen mus accuraely fi he imber srucure. Seel componens which prorude pas he surface of he wood should be avoided where possible. This reduces he ransmission of hea released by he fire inside he imber cross-secion. If greaer fire resisance is required, faseners can be fully proeced by cladding made of wooden maerials or mineral panel maerials. For furher srucural design informaion, Sora Enso refers he reader o [22], secion Double-layered componens Gaps beween double-layered componens mus be fully insulaed wih rock fibre. [34] (In he fire siuaion, his can, for insance, preven sparks or oher burning componens from falling beween double-layered walls. In addiion, by insulaing he caviies, a possible chimney effec which would favour he spread of fire can be prevened.) Page 35 of 50

Vehicle Arrival Models : Headway

Vehicle Arrival Models : Headway Chaper 12 Vehicle Arrival Models : Headway 12.1 Inroducion Modelling arrival of vehicle a secion of road is an imporan sep in raffic flow modelling. I has imporan applicaion in raffic flow simulaion where

More information

Reading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4.

Reading from Young & Freedman: For this topic, read sections 25.4 & 25.5, the introduction to chapter 26 and sections 26.1 to 26.2 & 26.4. PHY1 Elecriciy Topic 7 (Lecures 1 & 11) Elecric Circuis n his opic, we will cover: 1) Elecromoive Force (EMF) ) Series and parallel resisor combinaions 3) Kirchhoff s rules for circuis 4) Time dependence

More information

Structural Dynamics and Earthquake Engineering

Structural Dynamics and Earthquake Engineering Srucural Dynamics and Earhquae Engineering Course 1 Inroducion. Single degree of freedom sysems: Equaions of moion, problem saemen, soluion mehods. Course noes are available for download a hp://www.c.up.ro/users/aurelsraan/

More information

Electrical and current self-induction

Electrical and current self-induction Elecrical and curren self-inducion F. F. Mende hp://fmnauka.narod.ru/works.hml mende_fedor@mail.ru Absrac The aricle considers he self-inducance of reacive elemens. Elecrical self-inducion To he laws of

More information

MECHANICS OF MATERIALS Poisson s Ratio

MECHANICS OF MATERIALS Poisson s Ratio Poisson s Raio For a slender bar subjeced o axial loading: ε x x y 0 The elongaion in he x-direcion i is accompanied by a conracion in he oher direcions. Assuming ha he maerial is isoropic (no direcional

More information

Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle

Physics 235 Chapter 2. Chapter 2 Newtonian Mechanics Single Particle Chaper 2 Newonian Mechanics Single Paricle In his Chaper we will review wha Newon s laws of mechanics ell us abou he moion of a single paricle. Newon s laws are only valid in suiable reference frames,

More information

IB Physics Kinematics Worksheet

IB Physics Kinematics Worksheet IB Physics Kinemaics Workshee Wrie full soluions and noes for muliple choice answers. Do no use a calculaor for muliple choice answers. 1. Which of he following is a correc definiion of average acceleraion?

More information

Activity 4 Solutions: Transfer of Thermal Energy

Activity 4 Solutions: Transfer of Thermal Energy Aciviy 4 Soluions: Transfer of Thermal nergy 4.1 How Does Temperaure Differ from Thermal nergy? a) Temperaure Your insrucor will demonsrae molecular moion a differen emperaures. 1) Wha happens o molecular

More information

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n

Module 2 F c i k c s la l w a s o s f dif di fusi s o i n Module Fick s laws of diffusion Fick s laws of diffusion and hin film soluion Adolf Fick (1855) proposed: d J α d d d J (mole/m s) flu (m /s) diffusion coefficien and (mole/m 3 ) concenraion of ions, aoms

More information

WEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x

WEEK-3 Recitation PHYS 131. of the projectile s velocity remains constant throughout the motion, since the acceleration a x WEEK-3 Reciaion PHYS 131 Ch. 3: FOC 1, 3, 4, 6, 14. Problems 9, 37, 41 & 71 and Ch. 4: FOC 1, 3, 5, 8. Problems 3, 5 & 16. Feb 8, 018 Ch. 3: FOC 1, 3, 4, 6, 14. 1. (a) The horizonal componen of he projecile

More information

Lecture 4 Kinetics of a particle Part 3: Impulse and Momentum

Lecture 4 Kinetics of a particle Part 3: Impulse and Momentum MEE Engineering Mechanics II Lecure 4 Lecure 4 Kineics of a paricle Par 3: Impulse and Momenum Linear impulse and momenum Saring from he equaion of moion for a paricle of mass m which is subjeced o an

More information

1. VELOCITY AND ACCELERATION

1. VELOCITY AND ACCELERATION 1. VELOCITY AND ACCELERATION 1.1 Kinemaics Equaions s = u + 1 a and s = v 1 a s = 1 (u + v) v = u + as 1. Displacemen-Time Graph Gradien = speed 1.3 Velociy-Time Graph Gradien = acceleraion Area under

More information

LINEAR SLOT DIFFUSERS

LINEAR SLOT DIFFUSERS Supply, Reurn, Dummy Linear Slo Diffusers A OSLD OTIMA model OSLD is a supply linear slo diffuser wih inegral volume conrol damper and hi and miss air sraighening deflecors. h Hi and miss air sraigheners

More information

The equation to any straight line can be expressed in the form:

The equation to any straight line can be expressed in the form: Sring Graphs Par 1 Answers 1 TI-Nspire Invesigaion Suden min Aims Deermine a series of equaions of sraigh lines o form a paern similar o ha formed by he cables on he Jerusalem Chords Bridge. Deermine he

More information

At the end of this lesson, the students should be able to understand

At the end of this lesson, the students should be able to understand Insrucional Objecives A he end of his lesson, he sudens should be able o undersand Sress concenraion and he facors responsible. Deerminaion of sress concenraion facor; experimenal and heoreical mehods.

More information

Sub Module 2.6. Measurement of transient temperature

Sub Module 2.6. Measurement of transient temperature Mechanical Measuremens Prof. S.P.Venkaeshan Sub Module 2.6 Measuremen of ransien emperaure Many processes of engineering relevance involve variaions wih respec o ime. The sysem properies like emperaure,

More information

In this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should

In this chapter the model of free motion under gravity is extended to objects projected at an angle. When you have completed it, you should Cambridge Universiy Press 978--36-60033-7 Cambridge Inernaional AS and A Level Mahemaics: Mechanics Coursebook Excerp More Informaion Chaper The moion of projeciles In his chaper he model of free moion

More information

Micrologic Control units 2.0 and 5.0 Low Voltage Products

Micrologic Control units 2.0 and 5.0 Low Voltage Products Micrologic Conol unis.0 and 5.0 Low Volage Producs User manual We do more wih eleciciy. Micrologic Conol unis.0 and 5.0 Discovering your conol uni Idenifying your conol uni Overview of funcions 4 Seing

More information

ANALYSIS OF REINFORCED CONCRETE BUILDINGS IN FIRE

ANALYSIS OF REINFORCED CONCRETE BUILDINGS IN FIRE ANALYSIS OF REINFORCED CONCRETE BUILDINGS IN FIRE Dr Zhaohui Huang Universiy of Sheffield 6 May 2005 1 VULCAN layered slab elemens: connecion o beam elemens Plae Elemen Slab nodes y x Reference Plane h

More information

Heat Transfer. Revision Examples

Heat Transfer. Revision Examples Hea Transfer Revision Examples Hea ransfer: energy ranspor because of a emperaure difference. Thermal energy is ransferred from one region o anoher. Hea ranspor is he same phenomena lie mass ransfer, momenum

More information

PHYS 1401 General Physics I Test 3 Review Questions

PHYS 1401 General Physics I Test 3 Review Questions PHYS 1401 General Physics I Tes 3 Review Quesions Ch. 7 1. A 6500 kg railroad car moving a 4.0 m/s couples wih a second 7500 kg car iniially a res. a) Skech before and afer picures of he siuaion. b) Wha

More information

23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes

23.2. Representing Periodic Functions by Fourier Series. Introduction. Prerequisites. Learning Outcomes Represening Periodic Funcions by Fourier Series 3. Inroducion In his Secion we show how a periodic funcion can be expressed as a series of sines and cosines. We begin by obaining some sandard inegrals

More information

Econ107 Applied Econometrics Topic 7: Multicollinearity (Studenmund, Chapter 8)

Econ107 Applied Econometrics Topic 7: Multicollinearity (Studenmund, Chapter 8) I. Definiions and Problems A. Perfec Mulicollineariy Econ7 Applied Economerics Topic 7: Mulicollineariy (Sudenmund, Chaper 8) Definiion: Perfec mulicollineariy exiss in a following K-variable regression

More information

Development of a new metrological model for measuring of the water surface evaporation Tovmach L. Tovmach Yr. Abstract Introduction

Development of a new metrological model for measuring of the water surface evaporation Tovmach L. Tovmach Yr. Abstract Introduction Developmen of a new merological model for measuring of he waer surface evaporaion Tovmach L. Tovmach Yr. Sae Hydrological Insiue 23 Second Line, 199053 S. Peersburg, Russian Federaion Telephone (812) 323

More information

V AK (t) I T (t) I TRM. V AK( full area) (t) t t 1 Axial turn-on. Switching losses for Phase Control and Bi- Directionally Controlled Thyristors

V AK (t) I T (t) I TRM. V AK( full area) (t) t t 1 Axial turn-on. Switching losses for Phase Control and Bi- Directionally Controlled Thyristors Applicaion Noe Swiching losses for Phase Conrol and Bi- Direcionally Conrolled Thyrisors V AK () I T () Causing W on I TRM V AK( full area) () 1 Axial urn-on Plasma spread 2 Swiching losses for Phase Conrol

More information

23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes

23.5. Half-Range Series. Introduction. Prerequisites. Learning Outcomes Half-Range Series 2.5 Inroducion In his Secion we address he following problem: Can we find a Fourier series expansion of a funcion defined over a finie inerval? Of course we recognise ha such a funcion

More information

Solutions from Chapter 9.1 and 9.2

Solutions from Chapter 9.1 and 9.2 Soluions from Chaper 9 and 92 Secion 9 Problem # This basically boils down o an exercise in he chain rule from calculus We are looking for soluions of he form: u( x) = f( k x c) where k x R 3 and k is

More information

EECE251. Circuit Analysis I. Set 4: Capacitors, Inductors, and First-Order Linear Circuits

EECE251. Circuit Analysis I. Set 4: Capacitors, Inductors, and First-Order Linear Circuits EEE25 ircui Analysis I Se 4: apaciors, Inducors, and Firs-Order inear ircuis Shahriar Mirabbasi Deparmen of Elecrical and ompuer Engineering Universiy of Briish olumbia shahriar@ece.ubc.ca Overview Passive

More information

Final Spring 2007

Final Spring 2007 .615 Final Spring 7 Overview The purpose of he final exam is o calculae he MHD β limi in a high-bea oroidal okamak agains he dangerous n = 1 exernal ballooning-kink mode. Effecively, his corresponds o

More information

04. Kinetics of a second order reaction

04. Kinetics of a second order reaction 4. Kineics of a second order reacion Imporan conceps Reacion rae, reacion exen, reacion rae equaion, order of a reacion, firs-order reacions, second-order reacions, differenial and inegraed rae laws, Arrhenius

More information

Physics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs.

Physics 180A Fall 2008 Test points. Provide the best answer to the following questions and problems. Watch your sig figs. Physics 180A Fall 2008 Tes 1-120 poins Name Provide he bes answer o he following quesions and problems. Wach your sig figs. 1) The number of meaningful digis in a number is called he number of. When numbers

More information

t is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t...

t is a basis for the solution space to this system, then the matrix having these solutions as columns, t x 1 t, x 2 t,... x n t x 2 t... Mah 228- Fri Mar 24 5.6 Marix exponenials and linear sysems: The analogy beween firs order sysems of linear differenial equaions (Chaper 5) and scalar linear differenial equaions (Chaper ) is much sronger

More information

AP Chemistry--Chapter 12: Chemical Kinetics

AP Chemistry--Chapter 12: Chemical Kinetics AP Chemisry--Chaper 12: Chemical Kineics I. Reacion Raes A. The area of chemisry ha deals wih reacion raes, or how fas a reacion occurs, is called chemical kineics. B. The rae of reacion depends on he

More information

Curling Stress Equation for Transverse Joint Edge of a Concrete Pavement Slab Based on Finite-Element Method Analysis

Curling Stress Equation for Transverse Joint Edge of a Concrete Pavement Slab Based on Finite-Element Method Analysis TRANSPORTATION RESEARCH RECORD 155 35 Curling Sress Equaion for Transverse Join Edge of a Concree Pavemen Slab Based on Finie-Elemen Mehod Analysis TATSUO NISHIZAWA, TADASHI FUKUDA, SABURO MATSUNO, AND

More information

ANALYSES OF THE INTERFACE BETWEEN WALL ELEMENTS AND RENDERING LAYERS. Extended Abstract

ANALYSES OF THE INTERFACE BETWEEN WALL ELEMENTS AND RENDERING LAYERS. Extended Abstract INSTITUTO SUPERIOR TÉCNICO Universidade Técnica de Lisboa ANALYSES OF THE INTERFACE BETWEEN WALL ELEMENTS AND RENDERING LAYERS Exended Absrac Sara Maria Garcia Gaspar Ocober, 2011 1 INTRODUCTION Adhesion

More information

Scientific Herald of the Voronezh State University of Architecture and Civil Engineering. Construction and Architecture

Scientific Herald of the Voronezh State University of Architecture and Civil Engineering. Construction and Architecture Scienific Herald of he Voronezh Sae Universiy of Archiecure and Civil Engineering. Consrucion and Archiecure UDC 625.863.6:551.328 Voronezh Sae Universiy of Archiecure and Civil Engineering Ph. D. applican

More information

Traveling Waves. Chapter Introduction

Traveling Waves. Chapter Introduction Chaper 4 Traveling Waves 4.1 Inroducion To dae, we have considered oscillaions, i.e., periodic, ofen harmonic, variaions of a physical characerisic of a sysem. The sysem a one ime is indisinguishable from

More information

SPH3U: Projectiles. Recorder: Manager: Speaker:

SPH3U: Projectiles. Recorder: Manager: Speaker: SPH3U: Projeciles Now i s ime o use our new skills o analyze he moion of a golf ball ha was ossed hrough he air. Le s find ou wha is special abou he moion of a projecile. Recorder: Manager: Speaker: 0

More information

d 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3

d 1 = c 1 b 2 - b 1 c 2 d 2 = c 1 b 3 - b 1 c 3 and d = c b - b c c d = c b - b c c This process is coninued unil he nh row has been compleed. The complee array of coefficiens is riangular. Noe ha in developing he array an enire row may be divided or

More information

Calculation of the Two High Voltage Transmission Line Conductors Minimum Distance

Calculation of the Two High Voltage Transmission Line Conductors Minimum Distance World Journal of Engineering and Technology, 15, 3, 89-96 Published Online Ocober 15 in SciRes. hp://www.scirp.org/journal/wje hp://dx.doi.org/1.436/wje.15.33c14 Calculaion of he Two High Volage Transmission

More information

STATE-SPACE MODELLING. A mass balance across the tank gives:

STATE-SPACE MODELLING. A mass balance across the tank gives: B. Lennox and N.F. Thornhill, 9, Sae Space Modelling, IChemE Process Managemen and Conrol Subjec Group Newsleer STE-SPACE MODELLING Inroducion: Over he pas decade or so here has been an ever increasing

More information

ACE 562 Fall Lecture 5: The Simple Linear Regression Model: Sampling Properties of the Least Squares Estimators. by Professor Scott H.

ACE 562 Fall Lecture 5: The Simple Linear Regression Model: Sampling Properties of the Least Squares Estimators. by Professor Scott H. ACE 56 Fall 005 Lecure 5: he Simple Linear Regression Model: Sampling Properies of he Leas Squares Esimaors by Professor Sco H. Irwin Required Reading: Griffihs, Hill and Judge. "Inference in he Simple

More information

Linear Response Theory: The connection between QFT and experiments

Linear Response Theory: The connection between QFT and experiments Phys540.nb 39 3 Linear Response Theory: The connecion beween QFT and experimens 3.1. Basic conceps and ideas Q: How do we measure he conduciviy of a meal? A: we firs inroduce a weak elecric field E, and

More information

2. Nonlinear Conservation Law Equations

2. Nonlinear Conservation Law Equations . Nonlinear Conservaion Law Equaions One of he clear lessons learned over recen years in sudying nonlinear parial differenial equaions is ha i is generally no wise o ry o aack a general class of nonlinear

More information

Chapter 14 Homework Answers

Chapter 14 Homework Answers 4. Suden responses will vary. (a) combusion of gasoline (b) cooking an egg in boiling waer (c) curing of cemen Chaper 4 Homework Answers 4. A collision beween only wo molecules is much more probable han

More information

Class Meeting # 10: Introduction to the Wave Equation

Class Meeting # 10: Introduction to the Wave Equation MATH 8.5 COURSE NOTES - CLASS MEETING # 0 8.5 Inroducion o PDEs, Fall 0 Professor: Jared Speck Class Meeing # 0: Inroducion o he Wave Equaion. Wha is he wave equaion? The sandard wave equaion for a funcion

More information

Predator - Prey Model Trajectories and the nonlinear conservation law

Predator - Prey Model Trajectories and the nonlinear conservation law Predaor - Prey Model Trajecories and he nonlinear conservaion law James K. Peerson Deparmen of Biological Sciences and Deparmen of Mahemaical Sciences Clemson Universiy Ocober 28, 213 Ouline Drawing Trajecories

More information

Methodology. -ratios are biased and that the appropriate critical values have to be increased by an amount. that depends on the sample size.

Methodology. -ratios are biased and that the appropriate critical values have to be increased by an amount. that depends on the sample size. Mehodology. Uni Roo Tess A ime series is inegraed when i has a mean revering propery and a finie variance. I is only emporarily ou of equilibrium and is called saionary in I(0). However a ime series ha

More information

Keywords: thermal stress; thermal fatigue; inverse analysis; heat conduction; regularization

Keywords: thermal stress; thermal fatigue; inverse analysis; heat conduction; regularization Proceedings Inverse Analysis for Esimaing Temperaure and Residual Sress Disribuions in a Pipe from Ouer Surface Temperaure Measuremen and Is Regularizaion Shiro Kubo * and Shoki Taguwa Deparmen of Mechanical

More information

Robotics I. April 11, The kinematics of a 3R spatial robot is specified by the Denavit-Hartenberg parameters in Tab. 1.

Robotics I. April 11, The kinematics of a 3R spatial robot is specified by the Denavit-Hartenberg parameters in Tab. 1. Roboics I April 11, 017 Exercise 1 he kinemaics of a 3R spaial robo is specified by he Denavi-Harenberg parameers in ab 1 i α i d i a i θ i 1 π/ L 1 0 1 0 0 L 3 0 0 L 3 3 able 1: able of DH parameers of

More information

Polymer Engineering (MM3POE)

Polymer Engineering (MM3POE) Polymer Engineering (MM3POE) VISCOELASTICITY hp://www.noingham.ac.uk/~eazacl/mm3poe Viscoelasiciy 1 Conens Wha is viscoelasiciy? Fundamenals Creep & creep recovery Sress relaxaion Modelling viscoelasic

More information

!!"#"$%&#'()!"#&'(*%)+,&',-)./0)1-*23)

!!#$%&#'()!#&'(*%)+,&',-)./0)1-*23) "#"$%&#'()"#&'(*%)+,&',-)./)1-*) #$%&'()*+,&',-.%,/)*+,-&1*#$)()5*6$+$%*,7&*-'-&1*(,-&*6&,7.$%$+*&%'(*8$&',-,%'-&1*(,-&*6&,79*(&,%: ;..,*&1$&$.$%&'()*1$$.,'&',-9*(&,%)?%*,('&5

More information

not to be republished NCERT MATHEMATICAL MODELLING Appendix 2 A.2.1 Introduction A.2.2 Why Mathematical Modelling?

not to be republished NCERT MATHEMATICAL MODELLING Appendix 2 A.2.1 Introduction A.2.2 Why Mathematical Modelling? 256 MATHEMATICS A.2.1 Inroducion In class XI, we have learn abou mahemaical modelling as an aemp o sudy some par (or form) of some real-life problems in mahemaical erms, i.e., he conversion of a physical

More information

Section 7.4 Modeling Changing Amplitude and Midline

Section 7.4 Modeling Changing Amplitude and Midline 488 Chaper 7 Secion 7.4 Modeling Changing Ampliude and Midline While sinusoidal funcions can model a variey of behaviors, i is ofen necessary o combine sinusoidal funcions wih linear and exponenial curves

More information

Example: Parametric fire curve for a fire compartment

Example: Parametric fire curve for a fire compartment Documen Ref: SX04a-EN-EU Shee 1 of 5 Tile Eurocode Ref EN 1991-1-:00 Made by Z Sokol Dae Jan 006 Checked by F Wald Dae Jan 006 Example: Parameric fire curve for a fire comparmen This example shows he deerminaion

More information

EE650R: Reliability Physics of Nanoelectronic Devices Lecture 9:

EE650R: Reliability Physics of Nanoelectronic Devices Lecture 9: EE65R: Reliabiliy Physics of anoelecronic Devices Lecure 9: Feaures of Time-Dependen BTI Degradaion Dae: Sep. 9, 6 Classnoe Lufe Siddique Review Animesh Daa 9. Background/Review: BTI is observed when he

More information

Lab #2: Kinematics in 1-Dimension

Lab #2: Kinematics in 1-Dimension Reading Assignmen: Chaper 2, Secions 2-1 hrough 2-8 Lab #2: Kinemaics in 1-Dimension Inroducion: The sudy of moion is broken ino wo main areas of sudy kinemaics and dynamics. Kinemaics is he descripion

More information

R t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t

R t. C t P t. + u t. C t = αp t + βr t + v t. + β + w t Exercise 7 C P = α + β R P + u C = αp + βr + v (a) (b) C R = α P R + β + w (c) Assumpions abou he disurbances u, v, w : Classical assumions on he disurbance of one of he equaions, eg. on (b): E(v v s P,

More information

Hall effect. Formulae :- 1) Hall coefficient RH = cm / Coulumb. 2) Magnetic induction BY 2

Hall effect. Formulae :- 1) Hall coefficient RH = cm / Coulumb. 2) Magnetic induction BY 2 Page of 6 all effec Aim :- ) To deermine he all coefficien (R ) ) To measure he unknown magneic field (B ) and o compare i wih ha measured by he Gaussmeer (B ). Apparaus :- ) Gauss meer wih probe ) Elecromagne

More information

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still.

Lecture 2-1 Kinematics in One Dimension Displacement, Velocity and Acceleration Everything in the world is moving. Nothing stays still. Lecure - Kinemaics in One Dimension Displacemen, Velociy and Acceleraion Everyhing in he world is moving. Nohing says sill. Moion occurs a all scales of he universe, saring from he moion of elecrons in

More information

Q.1 Define work and its unit?

Q.1 Define work and its unit? CHP # 6 ORK AND ENERGY Q.1 Define work and is uni? A. ORK I can be define as when we applied a force on a body and he body covers a disance in he direcion of force, hen we say ha work is done. I is a scalar

More information

Biol. 356 Lab 8. Mortality, Recruitment, and Migration Rates

Biol. 356 Lab 8. Mortality, Recruitment, and Migration Rates Biol. 356 Lab 8. Moraliy, Recruimen, and Migraion Raes (modified from Cox, 00, General Ecology Lab Manual, McGraw Hill) Las week we esimaed populaion size hrough several mehods. One assumpion of all hese

More information

10. State Space Methods

10. State Space Methods . Sae Space Mehods. Inroducion Sae space modelling was briefly inroduced in chaper. Here more coverage is provided of sae space mehods before some of heir uses in conrol sysem design are covered in he

More information

A First Course on Kinetics and Reaction Engineering. Class 19 on Unit 18

A First Course on Kinetics and Reaction Engineering. Class 19 on Unit 18 A Firs ourse on Kineics and Reacion Engineering lass 19 on Uni 18 Par I - hemical Reacions Par II - hemical Reacion Kineics Where We re Going Par III - hemical Reacion Engineering A. Ideal Reacors B. Perfecly

More information

Application Note AN Software release of SemiSel version 3.1. New semiconductor available. Temperature ripple at low inverter output frequencies

Application Note AN Software release of SemiSel version 3.1. New semiconductor available. Temperature ripple at low inverter output frequencies Applicaion Noe AN-8004 Revision: Issue Dae: Prepared by: 00 2008-05-21 Dr. Arend Winrich Ke y Words: SemiSel, Semiconducor Selecion, Loss Calculaion Sofware release of SemiSel version 3.1 New semiconducor

More information

T L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB

T L. t=1. Proof of Lemma 1. Using the marginal cost accounting in Equation(4) and standard arguments. t )+Π RB. t )+K 1(Q RB Elecronic Companion EC.1. Proofs of Technical Lemmas and Theorems LEMMA 1. Le C(RB) be he oal cos incurred by he RB policy. Then we have, T L E[C(RB)] 3 E[Z RB ]. (EC.1) Proof of Lemma 1. Using he marginal

More information

Learning Objectives: Practice designing and simulating digital circuits including flip flops Experience state machine design procedure

Learning Objectives: Practice designing and simulating digital circuits including flip flops Experience state machine design procedure Lab 4: Synchronous Sae Machine Design Summary: Design and implemen synchronous sae machine circuis and es hem wih simulaions in Cadence Viruoso. Learning Objecives: Pracice designing and simulaing digial

More information

Time series Decomposition method

Time series Decomposition method Time series Decomposiion mehod A ime series is described using a mulifacor model such as = f (rend, cyclical, seasonal, error) = f (T, C, S, e) Long- Iner-mediaed Seasonal Irregular erm erm effec, effec,

More information

Basic Circuit Elements Professor J R Lucas November 2001

Basic Circuit Elements Professor J R Lucas November 2001 Basic Circui Elemens - J ucas An elecrical circui is an inerconnecion of circui elemens. These circui elemens can be caegorised ino wo ypes, namely acive and passive elemens. Some Definiions/explanaions

More information

Morning Time: 1 hour 30 minutes Additional materials (enclosed):

Morning Time: 1 hour 30 minutes Additional materials (enclosed): ADVANCED GCE 78/0 MATHEMATICS (MEI) Differenial Equaions THURSDAY JANUARY 008 Morning Time: hour 30 minues Addiional maerials (enclosed): None Addiional maerials (required): Answer Bookle (8 pages) Graph

More information

CLASS XI SET A PHYSICS. 1. If and Let. The correct order of % error in. (a) (b) x = y > z (c) x < z < y (d) x > z < y

CLASS XI SET A PHYSICS. 1. If and Let. The correct order of % error in. (a) (b) x = y > z (c) x < z < y (d) x > z < y PHYSICS 1. If and Le. The correc order of % error in (a) (b) x = y > z x < z < y x > z < y. A hollow verical cylinder of radius r and heigh h has a smooh inernal surface. A small paricle is placed in conac

More information

Mathcad Lecture #8 In-class Worksheet Curve Fitting and Interpolation

Mathcad Lecture #8 In-class Worksheet Curve Fitting and Interpolation Mahcad Lecure #8 In-class Workshee Curve Fiing and Inerpolaion A he end of his lecure, you will be able o: explain he difference beween curve fiing and inerpolaion decide wheher curve fiing or inerpolaion

More information

Stability and Bifurcation in a Neural Network Model with Two Delays

Stability and Bifurcation in a Neural Network Model with Two Delays Inernaional Mahemaical Forum, Vol. 6, 11, no. 35, 175-1731 Sabiliy and Bifurcaion in a Neural Nework Model wih Two Delays GuangPing Hu and XiaoLing Li School of Mahemaics and Physics, Nanjing Universiy

More information

SUPPLEMENTARY INFORMATION

SUPPLEMENTARY INFORMATION SUPPLEMENTARY INFORMATION DOI: 0.038/NCLIMATE893 Temporal resoluion and DICE * Supplemenal Informaion Alex L. Maren and Sephen C. Newbold Naional Cener for Environmenal Economics, US Environmenal Proecion

More information

ACE 564 Spring Lecture 7. Extensions of The Multiple Regression Model: Dummy Independent Variables. by Professor Scott H.

ACE 564 Spring Lecture 7. Extensions of The Multiple Regression Model: Dummy Independent Variables. by Professor Scott H. ACE 564 Spring 2006 Lecure 7 Exensions of The Muliple Regression Model: Dumm Independen Variables b Professor Sco H. Irwin Readings: Griffihs, Hill and Judge. "Dumm Variables and Varing Coefficien Models

More information

DESIGN OF TENSION MEMBERS

DESIGN OF TENSION MEMBERS CHAPTER Srcral Seel Design LRFD Mehod DESIGN OF TENSION MEMBERS Third Ediion A. J. Clark School of Engineering Deparmen of Civil and Environmenal Engineering Par II Srcral Seel Design and Analysis 4 FALL

More information

Comprehensive Modelling Study of Quench Behaviour of the NHMFL 32 T All-Superconducting Magnet System. Input Data and Methodology Aspects.

Comprehensive Modelling Study of Quench Behaviour of the NHMFL 32 T All-Superconducting Magnet System. Input Data and Methodology Aspects. Comprehensive Modelling Sudy of Quench Behaviour of he NHMFL 32 T All-Superconducing Magne Sysem. Inpu Daa and Mehodology Aspecs. Andrew V. Gavrilin & Huberus W. Weijers Naional High Magneic Field Laboraory

More information

ODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004

ODEs II, Lecture 1: Homogeneous Linear Systems - I. Mike Raugh 1. March 8, 2004 ODEs II, Lecure : Homogeneous Linear Sysems - I Mike Raugh March 8, 4 Inroducion. In he firs lecure we discussed a sysem of linear ODEs for modeling he excreion of lead from he human body, saw how o ransform

More information

EXERCISES FOR SECTION 1.5

EXERCISES FOR SECTION 1.5 1.5 Exisence and Uniqueness of Soluions 43 20. 1 v c 21. 1 v c 1 2 4 6 8 10 1 2 2 4 6 8 10 Graph of approximae soluion obained using Euler s mehod wih = 0.1. Graph of approximae soluion obained using Euler

More information

- If one knows that a magnetic field has a symmetry, one may calculate the magnitude of B by use of Ampere s law: The integral of scalar product

- If one knows that a magnetic field has a symmetry, one may calculate the magnitude of B by use of Ampere s law: The integral of scalar product 11.1 APPCATON OF AMPEE S AW N SYMMETC MAGNETC FEDS - f one knows ha a magneic field has a symmery, one may calculae he magniude of by use of Ampere s law: The inegral of scalar produc Closed _ pah * d

More information

RC, RL and RLC circuits

RC, RL and RLC circuits Name Dae Time o Complee h m Parner Course/ Secion / Grade RC, RL and RLC circuis Inroducion In his experimen we will invesigae he behavior of circuis conaining combinaions of resisors, capaciors, and inducors.

More information

Sensing distance. 2 mm (slot width)

Sensing distance. 2 mm (slot width) Phoomicrosensor (Transmissive) EE-SX3 Ulra-Compac Slo / SMD Type (Slo widh: mm) PCB surface mouning ype. High resoluion wih a.3-mm-wide aperure. Be sure o read Safey Precauions on page 3. Ordering Informaion

More information

Designing Information Devices and Systems I Spring 2019 Lecture Notes Note 17

Designing Information Devices and Systems I Spring 2019 Lecture Notes Note 17 EES 16A Designing Informaion Devices and Sysems I Spring 019 Lecure Noes Noe 17 17.1 apaciive ouchscreen In he las noe, we saw ha a capacior consiss of wo pieces on conducive maerial separaed by a nonconducive

More information

Kriging Models Predicting Atrazine Concentrations in Surface Water Draining Agricultural Watersheds

Kriging Models Predicting Atrazine Concentrations in Surface Water Draining Agricultural Watersheds 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Kriging Models Predicing Arazine Concenraions in Surface Waer Draining Agriculural Waersheds Paul L. Mosquin, Jeremy Aldworh, Wenlin Chen Supplemenal Maerial Number

More information

Flow-Induced Vibration Analysis of Supported Pipes with a Crack

Flow-Induced Vibration Analysis of Supported Pipes with a Crack Flow-Induced Vibraion Analsis of Suppored Pipes wih a Crack Jin-Huk Lee, Samer Masoud Al-Said Deparmen of Mechanical Engineering American Universi of Sharjah, UAE Ouline Inroducion and Moivaion Aeroacousicall

More information

2) Of the following questions, which ones are thermodynamic, rather than kinetic concepts?

2) Of the following questions, which ones are thermodynamic, rather than kinetic concepts? AP Chemisry Tes (Chaper 12) Muliple Choice (40%) 1) Which of he following is a kineic quaniy? A) Enhalpy B) Inernal Energy C) Gibb s free energy D) Enropy E) Rae of reacion 2) Of he following quesions,

More information

Robust estimation based on the first- and third-moment restrictions of the power transformation model

Robust estimation based on the first- and third-moment restrictions of the power transformation model h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 6 December 3 www.mssanz.org.au/modsim3 Robus esimaion based on he firs- and hird-momen resricions of he power ransformaion Nawaa,

More information

k 1 k 2 x (1) x 2 = k 1 x 1 = k 2 k 1 +k 2 x (2) x k series x (3) k 2 x 2 = k 1 k 2 = k 1+k 2 = 1 k k 2 k series

k 1 k 2 x (1) x 2 = k 1 x 1 = k 2 k 1 +k 2 x (2) x k series x (3) k 2 x 2 = k 1 k 2 = k 1+k 2 = 1 k k 2 k series Final Review A Puzzle... Consider wo massless springs wih spring consans k 1 and k and he same equilibrium lengh. 1. If hese springs ac on a mass m in parallel, hey would be equivalen o a single spring

More information

Turbulent Flows. Computational Modelling of Turbulent Flows. Overview. Turbulent Eddies and Scales

Turbulent Flows. Computational Modelling of Turbulent Flows. Overview. Turbulent Eddies and Scales School of Mechanical Aerospace and Civil Engineering Turbulen Flows As noed above, using he mehods described in earlier lecures, he Navier-Sokes equaions can be discreized and solved numerically on complex

More information

1 Nuclear particles and nuclear radiation may cause ionisation as they pass through matter.

1 Nuclear particles and nuclear radiation may cause ionisation as they pass through matter. 1 uclear paricles and nuclear radiaion may cause ionisaion as hey pass hrough maer. Which of he following is he mos ionising? A α paricles B β paricles C γ rays D neurons 2 An unsable nucleus recoils as

More information

On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature

On Measuring Pro-Poor Growth. 1. On Various Ways of Measuring Pro-Poor Growth: A Short Review of the Literature On Measuring Pro-Poor Growh 1. On Various Ways of Measuring Pro-Poor Growh: A Shor eview of he Lieraure During he pas en years or so here have been various suggesions concerning he way one should check

More information

Nature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time.

Nature Neuroscience: doi: /nn Supplementary Figure 1. Spike-count autocorrelations in time. Supplemenary Figure 1 Spike-coun auocorrelaions in ime. Normalized auocorrelaion marices are shown for each area in a daase. The marix shows he mean correlaion of he spike coun in each ime bin wih he spike

More information

control properties under both Gaussian and burst noise conditions. In the ~isappointing in comparison with convolutional code systems designed

control properties under both Gaussian and burst noise conditions. In the ~isappointing in comparison with convolutional code systems designed 535 SOFT-DECSON THRESHOLD DECODNG OF CONVOLUTONAL CODES R.M.F. Goodman*, B.Sc., Ph.D. W.H. Ng*, M.S.E.E. Sunnnary Exising majoriy-decision hreshold decoders have so far been limied o his paper a new mehod

More information

APPM 2360 Homework Solutions, Due June 10

APPM 2360 Homework Solutions, Due June 10 2.2.2: Find general soluions for he equaion APPM 2360 Homework Soluions, Due June 10 Soluion: Finding he inegraing facor, dy + 2y = 3e µ) = e 2) = e 2 Muliplying he differenial equaion by he inegraing

More information

MATHEMATICAL DESCRIPTION OF THEORETICAL METHODS OF RESERVE ECONOMY OF CONSIGNMENT STORES

MATHEMATICAL DESCRIPTION OF THEORETICAL METHODS OF RESERVE ECONOMY OF CONSIGNMENT STORES MAHEMAICAL DESCIPION OF HEOEICAL MEHODS OF ESEVE ECONOMY OF CONSIGNMEN SOES Péer elek, József Cselényi, György Demeer Universiy of Miskolc, Deparmen of Maerials Handling and Logisics Absrac: Opimizaion

More information

Chapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws

Chapter 15: Phenomena. Chapter 15 Chemical Kinetics. Reaction Rates. Reaction Rates R P. Reaction Rates. Rate Laws Chaper 5: Phenomena Phenomena: The reacion (aq) + B(aq) C(aq) was sudied a wo differen emperaures (98 K and 35 K). For each emperaure he reacion was sared by puing differen concenraions of he 3 species

More information

KINEMATICS IN ONE DIMENSION

KINEMATICS IN ONE DIMENSION KINEMATICS IN ONE DIMENSION PREVIEW Kinemaics is he sudy of how hings move how far (disance and displacemen), how fas (speed and velociy), and how fas ha how fas changes (acceleraion). We say ha an objec

More information

Comparing Means: t-tests for One Sample & Two Related Samples

Comparing Means: t-tests for One Sample & Two Related Samples Comparing Means: -Tess for One Sample & Two Relaed Samples Using he z-tes: Assumpions -Tess for One Sample & Two Relaed Samples The z-es (of a sample mean agains a populaion mean) is based on he assumpion

More information

Exam 1 Solutions. 1 Question 1. February 10, Part (A) 1.2 Part (B) To find equilibrium solutions, set P (t) = C = dp

Exam 1 Solutions. 1 Question 1. February 10, Part (A) 1.2 Part (B) To find equilibrium solutions, set P (t) = C = dp Exam Soluions Februar 0, 05 Quesion. Par (A) To find equilibrium soluions, se P () = C = = 0. This implies: = P ( P ) P = P P P = P P = P ( + P ) = 0 The equilibrium soluion are hus P () = 0 and P () =..

More information

Mathcad Lecture #7 In-class Worksheet "Smart" Solve Block Techniques Handout

Mathcad Lecture #7 In-class Worksheet Smart Solve Block Techniques Handout Mahcad Lecure #7 In-class Workshee "Smar" Solve Block echniques Handou A he end of his lecure, you will be able o: use funcions in solve block equaions o improve convergence consruc solve blocks wih minimal

More information