The JCSS Probabilistic Model Code for Timber Examples and Discussion

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1 The JCSS Probablsc Model Code for Tmber Examples and Dscusson Jochen Köhler Research Assocae Mchael Faber Professor Insue of Srucural Engneerng Swss Federal Insue of Technology Zurch, Swzerland 1. Inroducon Durng he las decades srucural relably mehods have been furher developed, refned and adaped and are now a a sage where hey are beng appled n praccal engneerng problems as a decson suppor ool n connecon wh desgn and assessmen of srucures. Furhermore, basc knowledge concernng he acons on srucures and he maeral characerscs has mproved due o ncreased focus, beer measurng echnques and nernaonal research co-operaon. Ths knowledge has now reached a level where enables expers o ake no accoun unceranes n maeral properes and acons when assessng he load carryng capacy, servceably and servce lfe of srucures. Ths s no leas due o he fundamenal works on srucural relably mehods performed whn he Jon Commee on Srucural Safey (JCSS) ncludng, among ohers, he basc repors on acons on srucures, basc repors on maeral ressances and he gudelne for relably based assessmen of srucures. These documens provde general gudelnes for he use of srucural relably mehods n praccal applcaons and a he same me hese documens consue he bass for ensurng ha such analyss are performed on a heorecally conssen and comparave bass. The recen renforcemens n provdng such a bass for desgn are condensed n he almos complee JCSS Probablsc Model Code (PMC) [1]. In lne wh he ongong jon effors o complee he JCSS Probablsc Model Code, a chaper abou he probablsc modellng of mber maeral properes has been added recenly. The proposed probablsc model for mber maeral properes s srucured no several levels of sophscaon. The basc level reflecs he recen pracce for relably based code calbraon. The bendng srengh and sffness and he densy of mber are referred o as reference maeral properes and are nroduced as smple random varables. Furhermore, several possble refnemens are proposed. New nformaon mgh be nroduced, and s shown how dfferen ypes of new nformaon can be negraed by usng a Bayesan updang scheme. Refnemens n regard o he modellng of damage as a consequence of me load duraon are proposed. For he bendng srengh, a herarchcal spaal varably model s proposed and a mehod s presened for lnkng he properes of a cross secon (whch s consdered as he reference sarng pon for he modellng of spaal varably) wh he properes of a es specmen. In hs paper, several examples demonsrae he applcably of he probablsc model code for dfferen problems n mber research, engneerng and code wrng. General reference s made o he JCSS PMC and he neresed reader s nved o oban hs documen from JCSS [1].

2 2. Applcaons and Examples 2.1. Relably Based Code Calbraon In Faber and Sorensen [2] he prncples of relably based code calbraon s demonsraed. In he followng he approach followed here s llusraed along wh an example on code calbraon for a mber desgn code. Relably analyss of srucures for he purpose of code calbraon n general or for he relably verfcaon of specfc srucures requres ha he relevan falure modes be represened n erms of lm sae funcons. The lm sae funcons defne he realzaons of ressance parameers,.e. he maeral properes and he load varables resulng n srucural falure. In code based desgn formas such as he Eurocodes [3], desgn equaons are prescrbed for he verfcaon of he capacy of dfferen ypes of srucural componens n regard o dfferen modes of falure. The ypcal forma for he verfcaon of a srucural mber componen n Eurocode 5 [4] s gven as a desgn equaon n he followng form: r g k z s = k [1] = k mod d ψγ s,, 0 γ M where r k s he characersc value for he ressance, zd s a vecor of desgn varables (e.g. he cross secon of a mber beam), s k, are he characersc values of load effecs whch are consdered n he desgn, γ M and γ s, are paral safey facors for he ressance and he loads respecvely. When more han one varable load s acng, load combnaon facors ψ are mulpled on one or more of he varable load componens o ake no accoun he fac ha s unlkely ha all varable loads are acng wh exreme values a he same me. k s a mod modfcaon facor akng no accoun he effec of he duraon of load and mosure. In hs example kmod s assumed o be uny,.e. no load duraon and mosure effecs are consdered. Accordng o Eq. [1] falure F corresponds o an even defned by F = { g 0}. The paral safey facors ogeher wh he characersc values are nroduced n order o ensure a ceran mnmum relably level for he srucural componens desgned accordng o desgn equaons as e.g. gven n Eq. [1]. As dfferen maerals have dfferen unceranes assocaed wh her maeral parameers he paral safey facors are n general dfferen for he dfferen maerals. In accordance wh a gven desgn equaon, such as e.g. Eq. [1], a relably analyss may be performed based on a lm sae funcon of smlar form as: G = z dxr S = 0 [2] where R and S are he ressance and he load effecs as random varables and X a he model uncerany. Wh gven probablsc models for X, R and S he relably of a srucural mber componen desgned accordng o Eq. [1] wh a gven se of paral safey facors and characersc values can be checked by usng sandard procedures as e.g. FORM/SORM (see e.g. Madsen [5]). The am of relably based code calbraon s he calbraon of paral safey facors such ha he relably correspondng o dfferen ypcal desgn suaons are as close as possble o a specfed value for he arge relably. Recommendaons for suable arge relables are provded by e.g. he JCSS [1] or ISO [6]. Dfferen desgn suaon mgh be consdered e.g. hrough dfferen conrbuons of dfferen load effecs, n he case of he lm sae funcons n Eqs. [1] and [2], and consderng wo load effecs due o permanen and varable load respecvely, hs mgh be nroduced hrough a facor α = L = 1,2,..., Las:

3 r g = z αγ s ( 1 α ) γ s = 0 α = L = 1,2,..., L [3] k d G G, k Q Q, k γ M ( 1 α ) G = zdxr αsg SQ = 0 α = L = 1,2,..., L [4] where s and Gk, sq, k are he characersc values of he load effecs due o permanen and varable load respecvely, S and S are he wo load effecs as random varables. G Q Accordngly, he followng opmzaon problem can be formulaed, [2]: L ( ) 2 ( ) w ( ) mnw γ = β γ β [5] γ = 1 β s he relably ndex correspondng o he arge relably, ( ) where β γ s he relably ndex correspondng o a desgn performed wh a se of paral safey facors γ and w, are = 1,2,..., L facors ndcang relave frequency / mporance of he dfferen desgn suaons. The code calbraon procedure as descrbed above s already ncluded n he sofware package CodeCal, a MS EXCEL based sofware, whch s provded as freeware by he JCSS [7]. In he T followng, he paral safey facors γ = ( γ M, γg, γq) are calbraed usng CodeCal. The calbraon akes bass n Eqs.[3]-[5]. The probablsc model of he random varables of he problem are chosen as proposed n he JCSS PMC [1] and shown n Tab. 1. Tab. 1 Parameers used n he example and resuls. From he JCSS PMC [1] Par. safey facors COV Ds. Type Quanle EC 5 Opmzed, β = 4.2 (yearly) Ressance 0.25 Lognormal 5% Model uncerany 0.05 Lognormal Effec of perm. load 0.1 Normal 50% Effec of var. load 0.4 Gumbel 98% L = 10 desgn suaons are consdered (see Eqs. [3]-[5]). Relably Index Fg. 1 Targe Calbraed EC 5 Relably ndex correspondng o dfferen desgn suaons. For a chosen arge relably ndex β = 4.2 (yearly, as recommended n he JCSS PMC [1]) he paral safey facors are calbraed by solvng Eq. [5]. The calbraed se of paral safey facors ogeher wh he paral safey facors prescrbed n EC 5 s gven n Tab. 1. In Fg. 1 he relably ndex β s ploed over he dfferen desgn suaons represened by dfferen values of α. I s shown ha he se of paral safey facors prescrbed n EC 5 [4] s no correspondng o he opmal se f all paral safey facors are subjec o calbraon. In load and ressance facor desgn (LRFD) formas as EC 5, however, paral safey facors for load effecs are n general smlar and ndependen from he srucural maeral whch s ulzed. Therefore, code calbraon procedures should nvolve all

4 possble buldng maerals,.e. calbrae paral safey facors consderng all relevan desgn suaons and maerals. The sofware package CodeCal faclaes hs opon. For he valdy of he resuls of a code calbraon procedure, s of umos mporance ha he basc random varables ulzed n he analyss are quanfed based on he bes knowledge avalable. The JCSS PMC [1] provdes a se of probablsc models, whch can be used f no oher nformaon abou he varables s avalable. E.g. for he mber bendng srengh, a lognormal dsrbuon wh COV = 0.25 s suggesed. However, f new nformaon (drec or ndrec) s avalable, he suggesed dsrbuon should be consdered as a pror dsrbuon n a Bayesan updang scheme (as descrbed n more deal n he JCSS PMC [1]) Probablsc Desgn of a Srucural Componen The probablsc desgn of a smple mber beam as llusraed n Fg. 2 s consdered n hs example. The arge relably ndex s specfed o β arge = 4.2 (yearly) and he hegh h of he beam should be desgned by gven wdh b and span l of he beam. I s known ha he mber of hs beam s graded o C30 (EN 338, [8]), whch corresponds o a characersc value for he bendng srengh r mk, = 30MPa and a mean value for he modulus of elascy of moe mmean, = MPa. I s also known ha a reasonable probablsc model of he load S s he ( S ) ( S ) Gumbel dsrbuon wh mean value Mean = 10 KN and sandard devaon SDev = 4 kn. The gven nformaon s summarzed n Tab. 2 and n Fg. 2. Tab. 2 Gven Informaon for he Probablsc Desgn Bendng Srengh (C30) Varable Load MOE (C30) Char. Value r mk, = 30MPa Mean Value moe mmean, = 12 GPa Rm Lognormal dsrbued, COV = 0.25 accordng PMC ( R m ) Mean = 46.4 MPa ( ) R m SDev S Gumbel dsrbued () S Mean = 10 kn () S SDev = 4 kn MOEm Lognormal dsrbued, COV = 0.13 accordng PMC ( MOE m ) Mean = 12 GPa ( MOE m ) SDev = 1.56 GPa b, h Fg. 2 Ss, l b= 150mm l = 6m Smple beam wh cenre load. The ulmae lm sae funcon s gven as: 2 G bh R Sl = / = 0 [6] The beam hegh whch corresponds o he arge relably ndex β arge = 4.2 (yearly) s calculaed as h = 293 mm. The sofware package SYSREL [9] s used and he frs order relably mehod (FORM) s choosen as he solvng opon Relably updang The relably of he beam desgned n hs example mgh be updaed by usng addonal nformaon. E.g. nformaon abou he beam sffness can be ulzed, as could be he resul of a md-span deflecon measuremen u m of he beam susanng a proof load s = 10 kn. The nformaon of he measuremen can be consdered by defnng a lm sae funcon as

5 3 = m48 m [7] H u MOE I sl where I s he momen of nera of he beam. If H = 0 he MOEm -value corresponds exacly o he suaon ha he deflecon s as measured. Relably Index Fg u [mm] m Updaed relably ndex over he correspondng deflecon measuremens. Assumng ha he mber modulus of elascy and he bendng srengh are correlaed, e.g. as gven n he JCSS PMC [1] wh ρ = 0.8, he falure probably P f can be updaed by solvng P G P = P G H = = f ( 0 0) where P( G 0 H 0) ( 0 H = 0) P( H = 0) [8] = s he probably of lm sae volaon, G 0, by gven md-span deflecon measuremen u m, H = 0. The sofware package SYSREL [9] s used o solve Eq. [8] for dfferen possble deflecon measuremens. The updaed relably ndex s ploed over dfferen possble measuremens n Fg. 3. Noe ha no model uncerany s consdered n Eqs. [6] and [7] for smplcy reasons Comparson of dfferen bendng es confguraons Tmber maeral properes are n general assessed by performng ess accordng o sandard es procedures whch are prescrbed n dfferen naonal and nernaonal codes. E.g. for he evaluaon of he bendng srengh, common es sandards are: n Europe he sandard EN 408 [10] ogeher wh EN 384 [11], n he Uned Saes s he ASTM D ogeher wh ASTM D [12] and n Ausrala / New Zealand s he sandard AS/NZ 4063:1992 [13]. In Tab. 3 he es specfcaons accordng o he dfferen sandards are llusraed. Whle he specfcaons n regard o he geomery and he clmae condonng of he es specmen s smlar, he prescrbed expeced me unl falure accordng o he Norh Amercan es sandard s 60 seconds, accordng o he European and Ausralan es sandard s 300 seconds. I can be expeced ha he srengh measuremens are sensve o he dfference of specfed me unl falure and he effec mgh be assessed by usng one of he DOL models proposed n he JCSS PMC [1]. Tab. 3 An overvew comparson beween dfferen bendng srengh es procedures. Orgn/Code Europe/EN 408 and EN 384 Norh Amerca/ ASTM D and Ausrala/New Zealand AS/NZ 4063:1992 Geomery 4 pon bendng L = 18 H H = 150 mm 4 pon bendng L = 17 H 21 H H = 150 mm 4 pon bendng L = 18 H H = 150 mm Clmae/Mosure Conen Condoned a Temp.: 20 C Rel. Hum.: 65% Mosure Conen: 13% Condoned a Temp.: 20 C Rel. Hum.: 65% Loadng/Tme o falure Ramp load, Tme o falure: 300 s ± 120 s Ramp load, Tme o falure: 60 s ~ (10s, 600s) Ramp load, Tme o falure: 300 s ± 120 s Bas (By judgmen) weakes secon n he mddle. Tenson sde random. (By judgmen) weakes secon whn suppors. Tenson sde random. In hs example s focused on he dfferen provsons for how o place a beam whn he suppors of he four pon bendng arrangemen (compare Fg. 4). -

6 Europe Uned Saes Ausrala Fg. 4 Illusraon of he effec of dfferen weak secon placng specfcaons; Europe (EN 384, weak secon beween load applcaon), Uned Saes (ASTM D , weak secon beween suppors), Ausrala (AS 4063:1992, weak secon a random). To llusrae he effec of he weak secon placng specfcaon, he model derved by Isaksson [14], s ulzed for smulaons. The model s used as s presened n he JCSS PMC [1] and he model parameers are quanfed as presened n Fg. 5. ln(bendng srengh) m, j ( v + ) R = exp + j [ MPa] ln(r ) j j ( ) v Normal 4.03;0.25 ( ) Normal 0;0.19 ( ) Normal 0;0.16 ( )[ mm] X Exponenal 1/480 x j longudnal drecon of he beam Fg. 5 Model for he spaal varaon of he bendng srengh and model parameers used n he presened example. A mber beam s consdered as a longudnal sequence of weak secons. In Fg. 5 v s he unknown logarhm of he mean srengh of all secons n all componens, ϖ s he dfference beween he logarhm of he mean srengh of he secons whn a componen and χj s he dfference beween he srengh weak secon j n he beam and ϖ. v, ϖ and χ j are modelled as ndependen normal dsrbued random varables. The Mone Carlo Smulaon echnque (see e.g. Melchers [15]) s used for he probably assessmen. I s assumed ha he beams have a lengh of 5 m bendng componens are smulaed wh a weak secon dsrbuon as ndcaed n Fg. 5. The componens are esed vrually accordng o he hree dfferen bendng srengh es specfcaons llusraed n Fg. 4. The obaned vrual srengh daa s ulzed o calbrae he parameers of a lognormal dsrbuon. The correspondng dsrbuon funcons are ploed ogeher wh he dsrbuon funcon of he

7 srengh of he weak secons n Fg. 6. The parameers of he dsrbuon funcons are also gven n Fg. 6. The dsrbuon funcons shown n Fg. 6 llusrae he dfference beween he bendng srengh of es specmen and he bendng srengh of all weak secons. As nroduced n he probablsc model code he bendng srengh of a weak secon corresponds o he bendng srengh of a reference volume. An neresng queson concernng he probablsc modellng of mber maeral properes s how o relae measuremens on es specmen rms, o he properes of reference volumes. For he gven example a smple relaon s found wh he form: r m,0 r ϑ m,0 rm, s = [9] The parameer ϑ s calbraed for dfferen es sandards by usng he leas squares echnque. The resuls are also gven n Fg. 6. Accordng o Eq. [9] he parameers ξ and δ of he lognormal dsrbuon can be relaed as: ξ ( ξr ) m s rm,0, ϑ = [10] ( r ) m s δ = δ ϑ rm,0, [11] 1 Dsrbuon parameers correspondng o he model descrbed above: 0.8 US Weak secons Probably Europe weak secon Europe Uned Saes Ausrala Ausrala Bendng Srengh [MPa] EN US AUS Fg. 6 Comparson of dsrbuon funcons and her lognormal parameers of smulaed bendng es specmen accordng o dfferen naonal es sandards. I should be underlned ha he gven example s based on he whn componen bendng srengh varaon model and he correspondng model parameers as presened n Isaksson [14] and JCSS PMC [1]. The resuls presened n hs example are also sensve o he assumed lengh of he es specmen. 3. Conclusons and Dscusson In he recen verson of he JCSS PMC [1] probablsc models for he basc mber maeral properes have been ncluded. The documen ncludes ndcave numercal values for he model

8 parameers and refnemens relaed o updang of he probablsc model gven new nformaon, spaal varaon of srengh and duraon of load effecs are descrbed. In he presen paper s demonsraed how he proposed model mgh be mplemened for ypcal problems n mber research, engneerng and code wrng. The JCSS PMC [1] can be seen as a gudelne and common reference for probably based code calbraon of mber desgn codes. The parameers of he proposed models, however, need o be quanfed on a broad and represenave daa base. Comprehensve expermenal daa concernng he basc mber phenomena already exs, especally resulng from research projecs n Norh Amerca, Europe and Ausrala. One major ask for developng furher he presened model code s o collec and assess exsng expermenal daa. The mber research communy s asked o conrbue by makng avalable expermenal daa for he quanfcaon of model parameers for mber predomnanly used for mber desgn. References 1. JCSS. Jon Commee on Srucural Safey - Probablsc Model Code. hp:// Faber M.H. and Sørensen J.D. Relably Based Code Calbraon - The JCSS Approach Proceedngs o he 9h Inernaonal Mechansms for Concree Srucures n Cvl Engneerng ICASP, San Francsco, USA. 3. Eurocode_0, EN 1990:2002 'Bass of Srucural Desgn'. 2002, European Commee for Sandardzaon (CEN). 4. Eurocode5, Eurocode 5 - ENV :2004 'Desgn of Tmber Srucures - Par 1-1: General'. 2004: European Commee for Sandardzaon (CEN). 5. Borg, M., Relable mber connecons. Progress n Srucural Engneerng and Maerals, (3): p ISO_2394, General Prncples on Relably for Srucures 1998, Inernaonal Organsaon for Sandardzaon. 7. JCSS, CodeCal - Relably Based Code Calbraon. hp:// EN_338, Srucural Tmber Srengh Classes. 2003, Comé Européen de Normalsaon, Brussels, Belgum. 9. SYSREL, Sucural Relably Analyss Program. 1997, RCP Consulng Sofware, hp:// 10. EN_408, European Sandard: Tmber srucures - Srucural Tmber - Deermnaon of some physcal and mechancal properes. 2004, Comé Européen de Normalsaon, Brussels, Belgum. 11. EN_384, Tmber Srucures; Srucural mber Deermnaon of characersc values of mechancal properes and densy. 2004, Comé Européen de Normalsaon, Brussels, Belgum. 12. ASTM, Book of Sandards Wood (Prn and CD-ROM). July AS/NZS, 4063:1992 Tmber - Sress-graded - In-grade srengh and sffness evaluaon Isaksson, T., Modellng he Varably of Bendng Srengh n Srucural Tmber, n Repor TVBK , Lund Insue of Technology. 15. Melchers, R.E., Srucural Relably Analyss and Predcon. Second Edon ed. 2002: John Wley & Sons. 437.

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