Nonlinear Analysis of the Extreme Wind Fragility of the Reactor Hall Frame

SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 20 | NUMBER: 1 | 2020 | JUNE © 2020 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 18 NONLINEAR ANALYSIS OF THE EXTREME WIND FRAGILITY OF THE REACTOR HALL FRAME Ju aj KRÁLIK1, Ju aj KRÁLIK, j . 2 1Depa men o S uc u al Mechanics, Facul y o Ci il Enginee ing, Slo ak Uni e si y o Technology in B a isla a, Radlinského 11, B a isla a, Slo akia 2Academy o Fine A s and Design in B a isla a, H iezdosla o o nám. 18, B a isla a, Slo akia ju aj.k alik@s uba.sk, [email p o ec ed] DOI: 10.35181/ ces-2020-0004 Abs ac . This pape desc ibe he me hodology and he esul s o he sa e y analysis o he Nuclea Powe Plan s uc u es unde impac o he ex eme clima ic loads. In he case o he Nuclea Powe Plan s uc u es, he design c i e ia a e s onge . The equi emen s o he in e na ional agency IAEA and NRC s anda ds a e based on he p obabili y o mean e u n pe iod equal o one pe 104 yea s. The agili y cu e o he ex eme wind was de e mined on he base o he nonlinea p obabili y analysis o he s eel hale ame conside ing ma e ial and geome ic nonlinea i y using ANSYS so wa e. The ailu e mode o he NPP s uc u es is exp essed by High Con idence o Load P obabili y o Failu e (HCLPF). Keywo ds NPP, Nonlinea , Sa e y, F agili y, P obabili y, Ex eme Wind, ANSYS. 1. In oduc ion This pape deals wi h he esis ance o he s eel hale ame o he nuclea powe plan (NPP) wi h eac o VVER440. These analyses a e based on he ecommenda ions o IAEA and US NRC [1-14], expe ience om simila analyses o NPPs ab oad [15, 16], new indings om p obabilis ic analyses o s uc u es and own expe ience om p e ious analyses [16]. The me hodology o p obabilis ic analysis o NPP s uc u es a e based on he analy ical solu ion and simula ion me hods. [17-34]. These analyses a e based on he cu en esul s o moni o ing he ma e ial p ope ies o he NPP s uc u es, as well as he esul s o he analyses o he esis ance o indi idual elemen s o he impo an s uc u es o he NPP objec s unde he in luence o a ious ypes o ini ia ing e en s. The in e na ional o ganiza ion IAEA in Vienna se up he design equi emen s o he sa e y and eliabili y o he NPP s uc u es. The ex eme en i onmen al e en s (e.g. wind, empe a u e, snow, explosion...) a e he impo an loads om he poin o he NPP sa e y pe o mance. The ex eme wind loads a e de ined wi h he p obabili y o mean e u n pe iod equal o one pe 104 yea s. This pape deals wi h he analysis o he s eel hale ame loaded wi h ex eme wind load. The IAEA [6-10] and NRC s anda ds [11-13] equi e se ing up he p obabili y o he s uc u e ailu e du ing he ex eme loads. Fig. 1: FEM model o he NPP buildings. Fig. 2: FEM model o he ex eme wind loads. SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 20 | NUMBER: 1 | 2020 | JUNE © 2020 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 19 The NPP wi h he eac o VVER440 consis ou buildings – eac o building, leng hwise side building, c oss side building and u bine hall. The FEM model (Fig. 1, 2) o he NPP was c ea ed om ollowing elemen s in so wa e ANSYS – BEAM188, SHELL181, SURF154 and MASS21. The model has 996.917 elemen s wi h 444.426 nodes and 2.666.556 DOF. Fig. 3: FEM model o he NPP u bine hale. The c i ical s eel ame o u bine hall s uc u e was in es iga ed (Fig. 3). The FEM model o he s eel hale ame consis he beam and mass elemen s o ANSYS p og am - BEAM188 and MASS21 (Fig. 4). Fig. 4: FEM model o he c i ical s eel hale ame. The s eel s uc u es a e made om s eel S235 wi h ollowing nominal ma e ial p ope ies [15, 35] y = 235MPa and u = 360MPa. (1) The median ma e ial p ope ies a e de ined wi h acco dance o Eu ocodes () ym y 12 σ =+ and ym y 1.2 = o 0.1 σ =, (2) () um u 12 σ =+ and um u 1.2 = o 0.1 σ =. The IAEA sa e y s anda ds [2, 3] de ines ou accep able le els o ailu e a e o s uc u es, sys ems and componen s o he NPP: A. High plas ic ailu e - la ge plas ic de o ma ion o s uc u es, sys ems and componen s wi h p obabili y o ailu e (collapse, c ack) equal o o g ea e han 10-1. The in es iga ion o he s a e o he in ingemen mus be ca ied ou by non-linea analysis. B. Medium plas ic ailu e - la ge plas ic de o ma ion o s uc u es, sys ems and componen s wi h p obabili y o ailu e (collapse, c ack) om 10-2 o 10-1. The in es iga ion o he s a e o he in ingemen mus be ca ied ou by non-linea analysis. Accep able equi emen s o enginee ing elemen designs a e de ined in ASME and o AISC building s uc u es. C. Small plas ic ailu e - limi ed plas ic de o ma ion o s uc u es, sys ems and componen s wi h p obabili y o ailu e (collapse, up u e) om 10-3 o 10-2. The in es iga ion o he s uc u al sa e y is implemen ed wi h acco dance o ASME's equi emen s. D. Wi hou plas ic ailu e - elas ic de o ma ion o s uc u es, sys ems and componen s wi h p obabili y o ailu e (collapse, c ack) equal in he ange o 10-4 o 10-3 o accep able s eel s eng h c i e ia 0.8 y o 1.2 y. Fo he p obabili y o ailu e om 10-5 o 10-4 accep able s eel s eng h c i e ia a e less han 0.8 y. The ex e nal e en classi ica ion, i applied, does no imply di e en load le els o he ex e nal e en scena ios and he e o e he design o i ems classi ied o ex e nal e en s should e e only o he ex eme alues o design basic ex e nal e en s, o o a combina ion o hese e en s whe e a leas one o hem is aken a , o close o, i s ex eme alue. 2. Loads and load combina ions The load combina ion o he de e minis ic calcula ion was conside ed acco ding o ENV 1990 [36] and IAEA equi emen s [6, 7] o he ul ima e limi s a e o he s uc u e as ollows: • De e minis ic me hod – ex eme design si ua ion Ed = Gd + Qd + WEd (3) whe e Gd is he design alue o he pe manen dead loads, Qd - he design alue o he pe manen li e loads, WEd - he design alue o he ex eme wind load (wi h p obabili y o ge ing highe alue o 10-4). In he case o p obabilis ic calcula ion and he ul ima e limi s a e o he s uc u e he load combina ion we ake ollowing: • P obabilis ic me hod – ex eme design si ua ion E = G + Q + WE = g a .Gm + q a .Qm + w a .WEm (4) whe e g a , q a , w a a e he a iable pa ame e s de ined in he o m o he his og am calib a ed o he load combina ion in compliance wi h Eu ocode and Gm, Qm and SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 20 | NUMBER: 1 | 2020 | JUNE © 2020 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 20 WEm a e he median alues o he pe manen dead loads, li e loads and ex eme wind loads. 3. P obabili y nonlinea assessmen The sa e y o he building s uc u es was de e mined by he sa e y unc ion SF in he o m [19, 20, 25 and 30] SF E R= and 01SF≤< (5) whe e E is he ac ion unc ion and R is he esis ance unc ion. The eliabili y unc ion RF is de ined in he o m () ,1 0RF g R E SF R E==−=−> (6) whe e () , g RE is he eliabili y unc ion. The p obabili y o ailu e can be de ined by he simple exp ession [ ] () 0PPREPRE=<= −<  (7) The eliabili y unc ion RF can be exp essed gene ally as a unc ion o he s ochas ic pa ame e s X1, X2 o Xn, used in he calcula ion o R and E. 12 n (, ,..., )RF g X X X= (8) The ailu e unc ion g({X}) ep esen s he condi ion (capaci y ma gin) o he eliabili y, which can be ei he an explici o implici unc ion o he s ochas ic pa ame e s and can be single (de ined on one c oss-sec ion) o complex (de ined on se e al c oss-sec ions, e.g., on a complex ini e elemen model). In he case o he nonlinea analysis, he co ec solu ion o he elas ic-plas ic beha io o he s uc u es is de e mined by he unc ion plas ici y. The HMH unc ion o he plas ici y was used o he nonlinea solu ion o he s eel echnology segmen s. This plas ici y unc ion is de ined in he o m y R = and e E σ =, (9) whe e he e ec i e s ess (Von Mises s ess). The ailu e o he s eel segmen s in he ame o he PSA analysis is de ined by he limi ed alues o he maximal s ain de o ma ion. This ailu e unc ion is de ined in he o m ay,θ R ε = and e E ε =, (10) whe e he e ec i e s ain (Von Mises s ain). The ailu e p obabili y is calcula ed om he e alua ion o he s a is ical pa ame e s and heo e ical model o he p obabili y dis ibu ion o he eliabili y unc ion Z = g(X) using he simula ion me hods. The ailu e p obabili y is de ined as he bes es ima ion on he base o nume ical simula ions in he o m () 1 10 N i i PIgX N= =≤    (11) whe e N in he numbe o simula ions, g(.) is he ailu e unc ion, I[.] is he unc ion wi h alue 1, i he condi ion in he squa e b acke is ul illed, o he wise is equal 0. 4. Wind load The load on a s uc u e due o he wind is depended on bo h wind eloci y and e ain oughness [37, 38]. The wind eloci y and he eloci y p essu e a e composed o a mean and a luc ua ing componen . The mean wind eloci y m should be de e mined om he basic wind eloci y b which depends on he wind clima e and he heigh a ia ion o he wind de e mined om he e ain oughness and o og aphy. The luc ua ing componen o he wind is ep esen ed by he u bulence in ensi y. Acco ding o STN EN 1991-1-4 [37] e ain co esponds o e ain ca ego y I, wi h he basic wind eloci y b.o = 24 m/s, which co espond o basic wind p essu e qb = 0,36 kNm-2 . The basic wind eloci y should be calcula ed acco ding o [37] as: b = cdi .cseason. b,0 = 1.1.24 = 24 m/s (12) whe e b is he basic wind eloci y, de ined as a unc ion o wind di ec ion and ime o yea a 10 m abo e g ound o e ain ca ego y I, b,0 is he undamen al alue o he basic wind eloci y, cdi is he di ec ional ac o , cseason is he season ac o . The oughness ac o c (z) accoun s o he a iabili y o he mean wind eloci y a he si e o he s uc u e due o he heigh abo e g ound le el and he g ound oughness o he e ain upwind o he s uc u e in he wind di ec ion conside ed and should be calcula ed acco ding o [37] as: c (z) = k .ln(z/z0) = 0.19.ln(30.85/0.05) = 1.007 (13) whe e z0 is he oughness leng h, k e ain ac o depending on he oughness leng h z0 calcula ed using k = 0.19(z0/z0,II)0.07 = 0.19(0.05/0.05)0.07 = 0.19 (14) whe e z0,II = 0.05m ( e ain ca ego y II), zmin is he minimum heigh , zmax is o be aken as 200m, unless o he wise speci ied in he NA, z0, zmin depend on he e ain ca ego y. Recommended alues a e gi en in [37] depending on i e ep esen a i e e ain ca ego ies. Whe e o og aphy (e.g. hills, cli s e c.) inc eases wind eloci ies by mo e han 5% he e ec s should be conside ed using he o og aphy ac o c0. The mean wind eloci y m(z) a a heigh z abo e he e ain depends on he e ain oughness and o og aphy and on he basic wind eloci y, b, and should be calcula ed acco ding o STN ENV 1991 [37] as: m(z) = c (z).c0(z), b = 1.007.1.24 = 24.16 m/s (15) SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 20 | NUMBER: 1 | 2020 | JUNE © 2020 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 21 whe e c (z) is he oughness ac o , c0(z) is he o og aphy ac o , aken as 1.0 unless o he wise speci ied. The in luence o neighbo ing s uc u es on he wind eloci y should be conside ed acco ding o STN ENV 1991 [37]. Ex e nal p essu e coe icien s o duo pi ch oo s we e aken acco ding o STN ENV 1991 [37] o examined ame, wi h his loca ion in middle o he hall, wi h angel o oo ≤ 5° o di e en wind di ec ions. The ne p essu e on a oo on he opposi e su aces was aken as a posi i e and di ec ed owa ds he su ace, p essu e coe icien o he in e nal p essu e cpi = + 0.2. The ex eme wind load (EWL) was de e mined om he maximum wind speed de e mined om he SHMU measu emen s [39] in he NPP locali y o he e u n pe iod 10-4 by yea and he p obabili y o non-exceedance 95% ewl.b = 53.9 m/s and p essu e pewl.b = 1.816kPa (16) The p essu e om he ex eme wind load is 4.98x highe han he design basic alue. To ex apola e he maximum quan i y o ain all om me eo ological measu emen esul s on he quan i y o ain all (measu ed in he ime pe iod om 12 o 48 hou s), o a mean ime o ecu ence 102 o 104 yea s, i is ecommended o use he Gumbel p obabili y dis ibu ion wi h he equi emen ha he p obabili y o he excess quan i y o ain all wi h mean ime epea 102, espec i ely 104 yea s, du ing he design o he powe plan li e ime is less han 0.5, espec i ely 0.005. The mean alues o he ex eme wind eloci y and wind p essu e a e ollowing ewl.m = 47.6 m/s and p essu e pewl.m = 1.437kPa (17) 5. Load Failu e E ec The ailu e e ec o he ex eme load on he yield s eng h py is de e mined o he selec ed c i ical elemen s o he s uc u e, elying on he esul s o de e minis ic analyses o he design alues o he load and s uc u al esis ance e ec s and he ailu e condi ion o he gi en s ess case. In he case o p obabilis ic analysis, he mean alues o he inpu a iables, he median alues o he s a ic o ces om he e ec s o he load Em and he co esponding esis ance alue o he s uc u e Rm. I we conside he load combina ion acco ding o he ela ions (1) and he s a ic quan i y om he e ec o he ex eme load Eewl o sepa a e om he o he e ec s o he o he load Eo, hen he ailu e e ec o he load py ge s in he ollowing o m yyewl . p p η =, (18) whe e he pa ame e ηy is de ined om he eliabili y unc ion RF (Eq.4 and 8). Fo he es ima ion o he ailu e (collapse o a pa o he s uc u e o he whole) i is also possible o use he philosophy o de e mina ion o he limi s a e by he pa ame e HCLPF (High con idence o low p obabili y o ailu e) [11, 15], which is used mainly o e alua ion o seismic esis ance o he s uc u e. Hence, HCLPF is de e mined by he lowes esis ance o an elemen o a gi en s uc u e as a whole and exp esses he ela i e esis ance o he elemen o a speci ied ex eme load exp essed by he ex eme load ailu e p essu e. This me hodology gi es a good "es ima e" o he s uc u al esis ance, bu does no accu a ely e lec he ma gin o esis ance o he elemen s o he combined s ess cases because s eng h condi ions a e non-linea in na u e. The pa ame e HCLPF is de e mined om he ela ionship HCLPF (CDFM) = kD. Py = kD. η y.Pewl, (19) whe e kD is he duc ili y ac o exp essing he capaci y o a s uc u al membe o sys em. 6. Fo mula ion o F agili y Cu es The p obabili y a ing is based on limi s a es. Limi s a es a e possible ypes o aul s in NPP objec unc ions. The he me ic zone is in e p e ed as an eme ging leak, which can be a small con olled leak o a la ge ca as ophic c ack o b eakage. The median load-bea ing capaci y o a gi en ype o diso de depends la gely on se e al ac o s, including ma e ial p ope ies, model assump ions, and ailu e c i e ia. These ac o s a e cha ac e ized by unce ain y and a iabili y, which mus be aken in o accoun in he p obabili y e alua ion o he p essu e iola ion. The p essu e ca ying capaci y o each ype o diso de is exp essed as a andom a iable wi h logno mal dis ibu ion when he unce ain y/ a iabili y is inco po a ed in o he o mula ion (by de ini ion, he andom a iable is logno mal di ided i i s na u al loga i hm is no mally dis ibu ed). In case o using analy ical me hods and FORM me hod o ailu e p obabili y e alua ion, p essu e esis ance can be exp essed only om wo pa ame e s - model and ma e ial unce ain y. The ailu e esis ance is desc ibed as: pu = pm.e a a (20) whe e pm is he median o he ailu e capaci y, e a is he loga i hmically dis ibu ed andom a iable wi h he uni median and loga i hmic s anda d de ia ion βE (i.e., Em = 1.0 and βE2 = Va [ln(E)], whe e ln(E) is he na u al loga i hm a is a logno mal dis ibu ion o a andom a iable wi h a uni a y median alue and a loga i hmic s anda d de ia ion o βR (i.e. Rm = 1.0 and βR2 = Va [ln(R)]), ep esen ing he unce ain y o ma e ial p ope ies. Using he na u al loga i hm o equa ion (20) we ge : ln(pu) = ln(pm) + ln(e a ) + ln( a ) (21) The i s exp ession on he igh is a cons an (de e minis ic) numbe , while he second and hi d exp essions a e no mally dis ibu ed andom a iables. Acco ding o he basic heo y o p obabili y, he sum o SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 20 | NUMBER: 1 | 2020 | JUNE © 2020 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 22 no mally dis ibu ed independen andom a iables is no mally di ided, wi h βC2 sca e ing equal o he sum o he de ia ions o andom a iable elemen s, i.e. in his case: βC2 = (βE2 + βR2) (22) Based on he assump ion ha andom modelling a iables and andom a iables o ma e ial p ope ies a e independen , βC can be conside ed as a "composi e" loga i hmic s anda d de ia ion. Thus, he ailu e capaci y o each ype o diso de is de ined in e ms o h ee pa ame e s: he median ailu e a e Pm, he loga i hmic s anda d de ia ion βE (co esponding o he andom a iable unce ain y o he load e ec E) and he loga i hmic s anda d de ia ion βR (co esponding o he andom a iable unce ain y o he s uc u al esis ance R). Sca e ing is due o a lack o knowledge o he di e ences be ween he analy ical model and he eal design as well as ma e ial p ope ies. Model inde e minacy is in luenced by simpli ying assump ions, model de ails, and hei abili y o cap u e gli ch condi ions on he ac ual s uc u e (also design e sus ac ual de ia ion). The unce ain y o he s eng h o he ma e ial lies in he lack o in o ma ion on i s p ope ies as well as hei a iabili y. Examples o sou ces o such unce ain ies include: a iabili y in conc e e s eng h, s eel slip limi , s ess-s ain ela ionship, s eel shell s eng h, and empe a u e in luence on ma e ial s eng h. I expe i- men al da a on ac ual ma e ial s eng h p ope ies a e a ailable, hey can be used o es ima e βR. In p ac ice, howe e , he de e mina ion o he βE and βR pa ame e s is la gely a ma e o good enginee ing es ima ion. In de e mining he ealis ic alues o β, expe ience om p e ious p obabilis ic s uc u al analyses, e.g. o e p essu e s udies, seismic isk p obabili y assessmen s, e c. and expe ience in de e mining he load- bea ing capaci y o a ious elemen s by es and analy ical me hods. F om a p ac ical poin o iew, he p obabili y o a alue o less han 90% is less han 5% when using a pa icula model / o mula. In his case βE = - ln (0.9) / 1.65 = 0.06 (23) The ac o 1.65 is due o he ac ha he alue o a no mally dis ibu ed andom a iable ha is exceeded wi h a p obabili y o 95% is he mean educed by 1.65 imes he s anda d de ia ion. I he gi en alue is exceeded wi h 84% p obabili y, exp (-0.006) = 0.94, i is 94% o he bes es ima e alue. The p obabili y o ailu e occu ing a a p essu e less han a speci ic alue o in e nal p essu e, pu, is exp essed in he case o logno mal dis ibu ion as ()() ewlu um P ob ln C ppp pp Φ β =≤=  (24) whe e p is p obabili y ailu e occu s a a p essu e less han o equal o pu, pewl is a andom p essu e capaci y, Φ (.) is a cumula i e dis ibu ion unc ion o a s anda d no mal andom a iable, ln(.) is a na u al loga i hm, pm is a median p essu e capaci y, β C is a loga i hmic s anda d de ia ion o p. The i s s ep o a wind p essu e e alua ion is he iden i ica ion o po en ial ailu e modes. Once he po en ial ailu e modes a e iden i ied, ailu e c i e ia a e o be es ablished om which he median capaci ies a e es ima ed. Fo each ailu e mode, he median capaci ies a e o be e alua ed by conduc ing independen limi s a e analyses using he speci ic ailu e c i e ia wi h he applied loading consis ing o wind p essu e and dead load. Along wi h he p essu e capaci ies o he leak ype ailu e modes, leak a eas a e o be es ima ed in a p obabilis ic manne . The expec ed leak a eas a e ailu e mode dependen . A e calcula ion o he agili y o condi ional p obabili y o ailu e o s uc u e a di e en loca ions we mus o conside a combina ion o wind p essu e induced ailu e p obabili ies o di e en b eak o leak loca ions wi hin building. Fig. 5: Family o F agili y Cu es Showing Modeling Unce ain y Tu bine hall may ail a di e en loca ions unde di e en ailu e modes (Fig. 5). Conside wo ailu e modes A and B, each wi h n - agili y cu es and espec i e p obabili ies pi (i=1,…,n) and qj (j=1,…,n). Then he union C = A∪B, he agili y FCij(x) is gi en by 𝐹󰇛𝑥󰇜=𝐹󰇛𝑥󰇜+𝐹󰇛𝑥󰇜−𝐹 󰇛𝑥󰇜∩𝐹 󰇛𝑥󰇜 (25) whe e he subsc ip s i and j indica e one o n - agili y cu es o he ailu e modes and x deno e a speci ic alue o he p essu e wi hin he con ainmen . The p obabili y pij associa ed wi h agili y cu e FCij(x) is gi en by pi. qj i he median capaci ies o he ailu e modes a e independen . The esul o he in e sec ion e m in (22) is FAj(x) .FBj(x) when he andomness in he ailu e mode capaci ies is independen and min[FAi(x), FBj(x)] when he ailu e modes a e pe ec ly dependen . The low is and he consequence o an acciden depends on he o al leak a ea. Mul iple leaks a di e en loca ions o he con ainmen (e.g. bellows, ha ch, and ai lock) may con ibu e o he o al leak a ea. Using he me hodology desc ibed abo e, we can ob ain he agili y cu es o leak a each loca ion. Fo a gi en acciden sequence, he induced acciden p essu e p obabili y dis ibu ion, h(x), is known. This is con ol ed wi h he agili y cu e o each leak loca ion o ob ain he p obabili y o leak om ha loca ion (PLi). I is unde s ood ha he e is no b eak o con ainmen up u e a his p essu e. SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 20 | NUMBER: 1 | 2020 | JUNE © 2020 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 23 𝑝 =ℎ󰇛𝑥󰇜󰇟1−𝐹 󰇛𝑥󰇜󰇠𝐹󰇛𝑥󰇜𝑑𝑥  , (26) He e he Fb(x) is he agili y o b eak a he loca ion and Fl(x) is he agili y o leak. The leak is o each loca ion is only speci ied as a andom a iable wi h a p obabili y dis ibu ion. This dis ibu ion o leak a ea is disc edi ed in o a numbe o anges wi h associa ed p obabili ies (aij, Pij). The e o e, he p obabili y o aij occu ing (i.e. loca ion i and a e ange j) is PLi*Pij . This is accomplished using he DPD logic. Al e na i ely, a Mon e Ca lo simula ion could be done by andomly sampling om each leak a e dis ibu ion ( om di e en loca ions) and summing hem. I we use simula ion me hods based on Mon e Ca lo o LHS me hodology, se e al ac o s a ec ing he ailu e o he s uc u al elemen can be aken in o accoun in he calcula ion o he ailu e p obabili y. Abo e all, i is a ac ha o he e ec s also condi ion he s uc u al elemen capaci y. The design o he NPP can be b oken a di e en loca ions, wi h di e en ypes o ex eme loads. I each ype o ailu e leads o a collapse o he building s uc u e o same p incipal s uc u al sys em, i is no necessa y o check mul iple ypes o ailu e. I he consequences o he ypes o ailu es a e he same, hen he p obabili y o ailu e could be calcula ed as he p obabili y o a "g oup" o se e al ypes o ailu es. 7. Nonlinea analysis The 2D model o he c i ical ame o he u bine hall s uc u es was c ea ed om a spa ial model by selec ing using he subs uc u e me hod [15]. The s i ness and weigh o he elemen s as well as he load co espond o he alues in he spa ial model. The model con ains 660 elemen s and 1074 nodes (Fig. 6). The model was es ed by linea calcula ion om i s own g a i y and shows he same maximum de lec ion alues. The limi s a e o he s eel ame was conside ed o u ilize he geome ic and ma e ial nonlinea i y in p og am ANSYS [15]. The geome ic nonlinea i y is based on he heo y o he la ge s ain, which is o en used o elas ic-plas ic elemen s. The elas ic-plas ic model o s eel ma e ial was aken in compliance wi h he Von Mises yield unc ion [39-42]. The New on-Raphson i e a ion me hod o sol e nonlinea equa ions was conside ed. The plas ici y model is de ined as shown on Fig. 7, as mul ilinea iso opic ha dening ma e ial model. The nonlinea analysis based on po en ial heo y conside ing he iso opic ma e ial p ope ies was made o he beam elemen s BEAM188 in he FEM model. The s eel is ypical iso opic ma e ial. The elas ic-plas ic beha io o he iso opic ma e ials is desc ibed by he Hube -Mises- Hencky (HMH) yield c i e ion. Fig. 6: Model o ex eme wind loads. Fig. 7: Elas o-plas ic S ess-S ain Cu e. Consequen ly he s ess-s ain ela ions a e ob ained om he ollowing ela ions {} [] {} {} () [] {} pl el el σεε ελ σ ∂  =−=−   ∂   Q dDdd Ddd o { } { } ep σε  = dDd (27) whe e ep   D is elas ic-plas ic ma ix in he o m [] [] [] [] ee ep e e T T QF DD DD F Q AD ∂∂ ∂σ ∂σ ∂∂ ∂σ ∂σ     =−    +    (28) The ha dening pa ame e A depends on he yield unc ion and model o ha dening (iso opic o kinema ic). HMH yield c i e ion is de ined in he o m () eq T σσκ = , (29) whe e eq σ is equi alen s ess in he poin and () T σκ is he yield s ess depends on he ha dening. SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 20 | NUMBER: 1 | 2020 | JUNE © 2020 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 24 Tab.1: Calcula ion o HCLPF pa ame e om nonlinea analysis on 2D ame o he u bine hall Limi Pa am. Loads HCLPF [kPa] s a e η inc emen 50% 95% y 4 0,35 2,542 1,828 u 4 0,62 4,576 3,290 The nonlinea analyze o he c i ical ames o NPP u bine hall objec s was pe o med by he New on- Raphson me hod in 74 i e a ion s eps o he 63 loads inc emen s. The o al load was conside ed o be a mul iple o he design load om ex eme wind o η = 4. Fig. 8: De o ma ion o he ame o he HCLPF=4.576 kPa. Fig. 9: Equi alen s ess o he ame o he HCLPF=4.576 kPa. The ailu e capaci y o ame on ex eme wind load is ob ained om plas ic calcula ion o ame [15]. The agili y cu e o he ex eme wind was calcula ed on he base o he nonlinea de e minis ic analysis o he s eel ame o median alues o inpu da a. The p obabili y o he s uc u e ailu e was calcula ed o he a ious le els o he wind loads. On he base o he nonlinea de e minis ic analysis he ac o o ailu e was calcula ed on he s eel hall ame. The Figs. 8-10 show esul s o 4x highe alue o he ex eme wind load. On base o he nonlinea de e minis ic calcula ion we ha e he median alue o he maximum wind load o elas ic limi e s a e ( y) and plas ic limi s a e ( u) (Tab.1) as ollows py.m = 2.542kPa and pu.m = 4.576kPa (30) Fig. 10: Maximal capaci y o ex eme wind. The duc ili y ac o can be de e mined as he a io be ween he plas ic limi load and elas ic limi s a e Duy 2.48/1.40 1.77k η η == = (31) 8. Unce ain ies o inpu da a The unce ain ies o he inpu da a – ac ion e ec and esis ance a e o he case o he p obabilis ic calcula ion o he s uc u e eliabili y de ined in JCSS [44] and Eu ocode 1990 [36]. Tab.2: P obabilis ic model o inpu pa ame e s. Name Quan i y Cha ac . Value Va iable Pa ame . His og am Ma e ial Young’s Modulus E k e a No mal Load Dead G k g a No mal Li e Q k q a Gumbel Ex eme Wind W k w a Gumbel Resis ance S eel S eng h sk F k a Logno mal Model Ac ion Unce ain M E m e. a No mal Resis ance Unce . M R m . a No mal Tab.3: Cha ac e is ic inpu da a o he p obabilis ic model. Name Quan i y Mean S and. De ia i on Min. Value Max. Value Ma e ial Young’s Modulus 1 0.120 0.645 1.293 Load Dead 1 0.010 0.755 1.282 Li e 0.60 0.200 0 1 Ex eme Wind 0.30 0.150 0.500 1.032 Resis ance S eel S eng h sk 1 0.100 0.726 1.325 Model Ac ion Unce ain 1 0.100 0.875 1.135 Resis ance Unce . 1 0.100 0.875 1.135 The s i ness o he s uc u e is de e mined wi h he median alue o Young’s modulus Em and a iable ac o e a . Loads a e ep esen ed by hei s cha ac e is ic alues SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 20 | NUMBER: 1 | 2020 | JUNE © 2020 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 25 Gm, Qm, WE,m and a iable ac o s g a , q a and w a . The esis ance o he s eel is delimi ed by he cha ac e is ic alues o he s eng h ak and he a iable ac o a . The unce ain ies o he calcula ion model a e conside ed by a iable model ac o and a iable load ac o o Gauss‘s no mal dis ibu ion. The eliabili y analyses o he s uc u es a e di e en ia ed om he poin o iew o design quan i ies as he nonlinea de e minis ic analyses [24]. In s ochas ic analysis, he mean alues and he s anda d de ia ion o he a iable quan i ies a e calcula ed analy ically o nume ically using Mon e Ca lo simula ion, which gi es us he mo e accu a e esul s han de e minis ic alues. Sensi i i y analysis calcula es he dominan impac o he ou pu quan i ies and he p obabili y o he ailu e is de ined in compa ison wi h he simula ed quan i ies in p obabilis ic analysis. 9. Wind agili y cu e o ame The p obabili y o he s eel ame ailu e was de e mined by he p obabilis ic analysis based on nonlinea analysis using he analy ical and simula ion me hods. The agili y cu e was calcula ed o a ious le els o wind loads using he esul s om he nonlinea analysis o he s eel hall ame. The p obabili y o ame ailu e was de e mined by wo me hods: A. Analy ical analysis based on he FORM me hod and conside ing he logno mal dis ibu ion o ac ion e ec E and esis ance R, B. LHS simula ion me hods in so wa e FREeT [31], conside ing he dis ibu ion o Gumbel's wind load, no mal sel -weigh dis ibu ion, and logno mal o esis ance (see Tab. 2-3). A) FORM es ima ion o ailu e p obabili y: We conside he median alue o he load limi e ec pwm = 4.576 MPa acco ding o Tab. 1, he loga i hmic s anda d de ia ion o load a e conside ed by alues β E = 0.1 and esis ance β R = 0.1. We ha e ollowing 22 CER 0.141 βββ =+= and (32) EWLN.95 wm C exp( 1.65 ) 3.133HCLPF p β =−= kPa In he case o ex eme wind loads, i is an ex eme load in he ange o 10 min. impac on s uc u e. In his case i is possible o conside he plas ic ese e o he uss gi de , o beam espec i ely. In he case o he simple console o beam simply suppo ed he duc ili y ac o is equi alen o kD = 1.5. Then we ge he alue o he limi load capaci y calcula ed om he linea analysis conside ing he duc ili y ac o EWLD.95 D wm C exp( 1.65 ) 1.836HCLPF k p kPa β =−= (33) B) LHS simula ion using he so wa e FREeT [31] The calcula ion based on LHS simula ion me hod (Figs. 11-13) is based on he No mal load dis ibu ion o he sel - weigh , he dis ibu ion o he ex eme wind load in he o m o a Gumbel dis ibu ion, and he Logno mal dis ibu ion o he s uc u al esis ance. The HCLPF pa ame e de e mined by he p obabili y analysis o he u bine hall ame unde ex eme wind load by LHS simula ion me hod o 1000 simula ions was ob ained as ollows () EWLD.95 1.8191HCLPF LHS kPa= (34) The wind agili y cu e o he s eel hall ame is p esen ed in Fig. 13. This cu e was calcula ed by LHS me hod using he p og am FREET. Fig. 11: His og am o he ex eme wind load. Fig. 12: His og am o HCLPF pa ame e o ex eme wind load. Fig. 13: Wind agili y cu e o he s eel hall ame. 10. Conclusion This pape p esen s he eliabili y analysis o he s eel hall ame esis ance due o ex eme wind loads [30]. The ex eme loads we e de ined o mean e u n pe iod equal SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 20 | NUMBER: 1 | 2020 | JUNE © 2020 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 26 o one pe 104 yea s in acco dance o he IAEA equi emen s o NPP s uc u es [5 o 10]. The geome ic and ma e ial nonlinea i y we e aken in o accoun . The de e minis ic and p obabilis ic analysis o he s uc u e ailu e was in es iga ed. The limi s a e ( ame collapse) was ob ained om de e minis ic analysis o he ac o ηu=2.48. The p obabili y o ailu e was calcula ed on p og am FREET using LHS me hod [31]. The p obabili y o ailu e alue is lowe han 10-6. In he case o he wind load mul iplied by ac o ηu = 2.48 he p obabili y o ailu e is equal o 0.0087. The wind agili y cu e o he s eel hall ame was de e mined using LHS me hod o a ious le el o he wind load. The lowes ailu e load alues (HCLPF) assuming a 95% p obabili y o no exceeding he ul ima e limi c i e ion on NPP s uc u al elemen s based on linea analysis a e as ollows: • Ex eme wind: HCLPFEWL.95 = 1.224 kPa Conside ing he minimum duc ili y alue o kD = 1.5 based on he assump ion o he plas ic ese e co esponding o he s a ic unce ain y o he ame sys ems, wi h he sho - e m exposu e o ex eme loads, we ha e he ollowing alues: • Ex eme wind: HCLPFEWLD.95 = 1.819 kPa Lowes Failu e Loads (HCLPF) assuming 95% p obabili y o no exceeding he s uc u al ailu e c i e ion o c i ical ames wi h duo-pi ch P a T uss and beam based on nonlinea analysis a e as ollows: • Ex eme wind: HCLPFEWLN.95 = 3.133 kPa The esul s o his analysis clea ly con i m ha u bine hall s uc u es ha e signi ican ly g ea e esis ance o ex eme snow and ex eme wind e ec s a he o igin o and he de elopmen o plas ic de o ma ions, conside ing he conse a i e linea calcula ion and assessmen o one (weakes ) c oss-sec ion o he c i ical elemen . 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Vol. 859, pp. 114-121 (8 p), ISSN 1662-8985, DOI:10.4028/www.scien i ic.ne / AMR. 859.114.