Application of Semi-probabilistic Methods to Verification of Series System

SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 80 APPLICATION OF SEMI-PROBABILISTIC METHODS TO VERIFICATION OF SERIES SYSTEM Mi osla SÝKORA, Jana MARKOVÁ, Vi ali NADOLSKI Depa men o S uc u al Reliabili y, Klokne Ins i u e, Czech Technical Uni e si y in P ague, Šolíno a 7, P ague, Czech Republic mi osla .syko [email p o ec ed]; jana.ma ko a@c u .cz; i ali.nadolski@c u .cz DOI: 10.35181/ ces-2021-0018 Abs ac . Non-linea ini e elemen analysis (NLFEA) has become a widely used ool in he eliabili y e i ica ion o ein o ced conc e e (RC) s uc u es. Reliabili y assessmen o RC s uc u es is a challenging ask and simpli ied app oaches o eliabili y e i ica ions a e needed o allow o ou ine applica ions. Simpli ied semi-p obabilis ic me hods such as he pa ial ac o me hod (PFM) o he Me hod o Es ima ion o Coe icien o Va ia ion (ECoV) may yield o e - o unde -conse a i e app oxima ions. This con ibu ion in es iga es such e o s ela ed o he applica ions o hese wo me hods o se ies sys ems; he design esis ances ob ained by he p obabilis ic app oach a e conside ed as a e e ence le el. I appea s ha PFM p o ides good app oxima ions as he me hod is ocused on c i ical membe s in he cases unde conside a ion. ECoV may o e es ima e up o 20% in si ua ions whe e NLFEA based on mean and/ o cha ac e is ic alues ails o iden i y a domina ing ailu e mode. The maximum obse ed e o is a ibu ed mainly o he ailu e in iden i ying he ype o dis ibu ion o he sys em esis ance. The p esen ed limi ed analysis indica es se e al di ec ions o u he esea ch (analysis o o he ypes o s uc u al sys ems, cases wi h non-logno mal esis ances, e ec s o a numbe o sys em componen s, and hei mu ual co ela ions, and pe o mance o ad anced me hods o eliabili y e i ica ion). Recommenda ions on how o iden i y si ua ions when a signi ican e o can be expec ed should be p o ided o p ac ical applica ions o simpli ied semi-p obabilis ic me hods. Keywo ds NLFEA, ECoV, pa ial ac o me hod, ein o ced conc e e, eliabili y assessmen , se ies sys ems. 1. In oduc ion Non-linea ini e elemen analysis (NLFEA) has become a widely used ool in eliabili y e i ica ion o ein o ced conc e e (RC) s uc u es as he ad ancing enginee ing knowledge makes i possible o design and build mo e complex s uc u al sys ems. Howe e , he eliabili y assessmen o RC s uc u es is a challenging ask due o he g owing complexi y o nume ical s uc u al models, he la ge numbe o andom a iables, and small ailu e p obabili ies ela ed o Ul ima e Limi S a e e i ica ion. The use o pa ial ac o me hod can be a p agma ic and su icien app oach o many s uc u al sys ems. Howe e , he global sa e y concep o he ully p obabilis ic app oach seem o be mo e app op ia e in he cases wi h s ong nonlinea beha iou , he domina ing ole o ensile and ac u e mechanical p ope ies, o wi h mul iple ailu e modes o simila impo ance [1], [2], [3]. The ully p obabilis ic app oach makes i possible o ealis ically conside he andomness o inpu pa ame e s such as ma e ial, geome ical, and load cha ac e is ics. Howe e , such an app oach is gene ally ime-consuming and semi- p obabilis ic app oaches a e commonly applied in enginee ing p ac ice. To acili a e he ou ine applica ions o semi- p obabilis ic me hods (bu also o a ully p obabilis ic app oach), ope a ional ules a e being inco po a ed in o he p esen codes o p ac ice; o ins ance: • D a p EN 1990:2021 o he basis o design should include gene al p o isions o nonlinea analysis and a speci ic sec ion on he design assis ed by nume ical simula ions (such as ou comes o NLFEA). • Besides he gene al conc e e-speci ic ules o NLFEA, p EN 1992:2021 o he design and assessmen o conc e e s uc u es should include he in o ma i e Annex F “Nonlinea analyses p ocedu es”. This annex p o ides ope a ional guidance o applica ions o NLFEA along wi h he pa ial ac o me hod (PFM), he global ac o me hod (e.g., using he Me hod o Es ima ion o Coe icien o Va ia ion – ECoV) o he p obabilis ic app oach. P o isions on how o conside NLFEA- ela ed model unce ain ies will SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 81 be p o ided. • d a ib Model Code 2020, p o iding a backg ound o he conc e e s uc u es- ela ed p o isions in Eu ocodes, is aimed o gi e de ailed guidance ega ding he alida ion, quan i ica ion o modelling unce ain y, and applica ion o a ious eliabili y e i ica ion me hods including he PFM, ECoV, and p obabilis ic app oach. Simpli ied semi-p obabilis ic me hods such as PFM o ECoV a e commonly de ised o yield adequa e es ima es in mos p ac ically ele an applica ions (e.g. o design esis ance o a s uc u al sys em in his s udy). Ine i ably, o e - o unde -conse a i e app oxima ions migh be ob ained in some cases whe e he me hods a e oo simpli ied. A de ailed s udy o such disc epancies esul s in a wide ange o design si ua ions o p ac ical ele ance is he subjec o he esea ch p ojec suppo ed by he Czech Science Founda ion unde G an 20-01781S; selec ed esul s a e p esen ed in his con ibu ion. Fi s insigh s ega ding he pe o mance o PFM and ECoV ha e been ob ained by: • Ce enka J. e al. [4] who in es iga ed shea esis ance o a es ed beam whe e he minimum o he s i ups and conc e e con ibu ions ( hus a se ies sys em o wo componen s) we e conside ed. The s udy indica ed ha possibly signi ican e o s migh be expec ed o bo h me hods in pa icula si ua ions. Howe e , all obse a ions we e a he uzzy due o he domina ing e ec o model unce ain y. • Syko a e al. [5] ook he basis om [4] bu adop ed a mo e ealis ic model o shea esis ance whe e he s i ups and conc e e con ibu ions (~pa allel sys em) a e summed up. They obse ed insigni ican e o s, wi h bo h me hods p o iding sligh ly conse a i e es ima es. I seems ha he e i ica ion o se ies sys ems p esen s a challenge conce ning applica ions o PFM and ECoV. A se ies sys em is hus analysed he e. In con as o [4], he ollowing modi ica ions a e adop ed he e: • Model unce ain y—commonly ea ed sepa a ely, beyond applica ions o ECoV and possibly also o PFM—is no conside ed he e o ob ain clea e insigh s in o he pe o mance o he wo semi- p obabilis ic me hods. • The p obabilis ic models o a iables a e adop ed om he backg ound documen s o EN 1992-1- 1:2004 and p EN 1992-1-1:2021 so as hey well co espond o he pa ial ac o s o ma e ials. Consequen ly, o si ua ions wi h a single ailu e mode domina ing, he PFM and ECoV design esis ances well ma ch hose based on he p obabilis ic app oach, and he de iciencies o he semi-p obabilis ic me hods in he si ua ions wi h uly sys em beha iou can be well in es iga ed. The con ibu ion concludes wi h a discussion abou he limi a ions o he p esen ed analysis and he need o u he in es iga ions. 2. Semi-p obabilis ic Me hods Unde Conside a ion In NLFEA, he eliabili y condi ion is commonly o mula ed as: Ed ≤ Rd, (1) whe e Ed ep esen s he design alue o load e ec and Rd is he design alue o esis ance. This con ibu ion is ocused on es ima ing Rd by PFM and ECoV. Acco ding o PFM, design esis ance may be calcula ed by NLFEA using he design alues o ma e ial pa ame e s. In his s udy, he ollowing ela ionship is applied: Rd,PFM = Rmod( ck / γC; yk / γS), (2) whe e Rmod = model esis ance; ck = cha ac e is ic alue o conc e e comp essi e s eng h; γC = pa ial ac o o conc e e ( u he in o ma ion is in Sec ion 3); yk = cha ac e is ic alue o yield s eng h o s eel ein o cemen ; and γS = 1.15 – pa ial ac o o s eel. Based on he global ac o me hod, Če enka V. [3,4] p oposed an app oach sui able o ou ine NLFEA applica ions – ECoV. The me hod assumes ha he dis ibu ion o esis ance, R, can be desc ibed by a wo- pa ame e logno mal dis ibu ion [6], which is desc ibed by mean Rm and coe icien o a ia ion VR. The unde lying assump ion o logno mal esis ance is easonable o esis ances o many s uc u al sec ions, membe s, and pe haps also o sys ems. Acco ding o ECoV, he wo pa ame e s o a logno mal dis ibu ion can be es ima ed as: Rm ≈ Rmod( cm; ym), (3) VR,ECoV = ln(Rm / Rk) / 1.65, (4) whe e cm and ym = mean alues o conc e e comp essi e s eng h and yield s eng h o s eel ein o cemen , espec i ely; and Rk ≈ R mod( ck; yk) is he es ima e o a cha ac e is ic alue o esis ance (es ima e o a 5% ac ile o R). The design esis ance is hen es ima ed as: Rd,ECoV = Rm exp(-αR β VR,ECoV), (5) whe e αR = 0.8 is he sensi i i y ac o o a domina ing esis ance pa ame e ; and β = 3.8 is he a ge eliabili y index acco ding o EN 1990:2002 o Ul ima e Limi S a es (bo h o a e e ence pe iod o 50 yea s). In Eq. (4) and Eq. (5), he e ec o model unce ain y is in en ionally neglec ed as discussed abo e; see also [4]. No e ha he adop ed alues o αR and β imply ha he design alue o sys em esis ance is de e mined as a 1.18‰ ac ile o i s dis ibu ion. SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 82 ECoV sepa a es alea o y and epis emic unce ain ies and e lec s he dis inc na u e o di e en ailu e modes (yielding o ein o cemen , ailu e o conc e e in comp ession o ension, geome ical ins abili y) wi h only wo NLFEA uns o es ima e Rm and Rk. As a widely accep ed—simple and mos ly su icien ly accu a e— me hod, ECoV has been in oduced in ib MC 2010, d a ib MC 2020, and p EN 1992-1-1:2021. 3. Cases unde Conside a ion Two undamen al cases o wo-componen se ies sys ems a e conside ed in he ollowing analysis: 1. Case 1 wi h wo ailu e modes ac ing as a se ies sys em wi h logno mally dis ibu ed componen esis ances, 2. Case 2 is based on Case 1, bu assuming no mally dis ibu ed componen esis ances. Design alues ob ained by PFM and ECoV a e no malised o hose ob ained by he p obabilis ic app oach, Rd,p ob. Fo ins ance, when Rd,semi-p ob / Rd,p obab > 1, a semi-p obabilis ic me hod o e es ima es design esis ance, hus being on he unsa e side. As discussed abo e, model unce ain y is no conside ed in he nume ical analysis as i is ypically ea ed sepa a ely, beyond he applica ion o ECoV. No e ha he jus i ica ion o he alues o γC and γS acco ding o [7] indica es ha he model unce ain y ac o s ela ed o he ecommended alues in EN 1992-1-1:2004 a e e y close o uni y and hus he alues o he pa ial ac o s a e adop ed wi hou any adjus men wi h espec o model unce ain y (in en ionally igno ed in he ollowing analysis). 4. Se ies Sys em wi h Logno mal Componen Resis ances (Case 1) I is o en a gued ha simpli ied semi-p obabilis ic me hods may ail in cases wi h se e al local ex ema ha a e ypically caused by mul iple ailu e modes. To e i y his, Case 1 is ocused on a simple se ies sys em ha can well ep esen o ins ance shea esis ance o a conc e e membe ha is de e mined as he minimum o he conc e e and s i ups con ibu ions, Vc and Vs espec i ely: R = min(Vc, ρ Vs), (6) whe e ρ is a de e minis ic s udy pa ame e a bi a ily a ied dis ega ding p ac ical cons ain s such as bounds on ein o cemen a ios o de ailing ules. In enginee ing applica ions, i can ep esen a ein o cemen a io, ρ = As / Ac (wi h deno ing an a ea o s eel ein o cemen o o sec ion). Rela ionship (6) can hen be e-w i en as: R / Ac = min( c’, ρ s’). (7) He ea e , sys em esis ance based on (7) is e e ed o as R wi hou indica ing he no malisa ion wi h espec o Ac o simpli y he no a ion. Unce ain ies in he wo con ibu ions a e desc ibed by coe icien s o a ia ion o c’and s’ ha accoun o a iabili y o he espec i e s eng hs and geome ical a iables. Mu ually s a is ically independen con ibu ions a e desc ibed by logno mal dis ibu ions wi h he ollowing cha ac e is ics: • cm’ = 29.1 1MPa, V c’ = 21.3%, and c0.05’ = ck’ = 20.5 MPa, • ym’ = 489 MPa, V y’ = 10.1%, and y0.05’ = yk’ = 414 MPa. The coe icien s o a ia ion a e de e mined in such a way ha he design alue o he conc e e and ein o cemen s eng hs co espond o he alue de e mined by PFM: cd’ = cm’ exp(-αR β V c’ ) = ck’ / 1.35, (8) yd’ = ym’ exp(-αR β V y’ ) = yk’ / 1.15. (9) The alue o γC is educed as he common alue o 1.5 co e s addi ional ac o o 1.15 o accoun o unce ain y a ising om conc e e being es ed on pu pose-made specimens in a lab, a he han in he inished s uc u e [7]. I mus be emphasised ha he adop ed alues o CoVs a e cha ac e is ic a he o esis ances han o s eng hs; his is why he symbol “ ’ ” is used in ela ionships om (7) o (9). Fig. 1 displays a iabili y o Rd,PFM / Rd,p obab and Rd,ECoV / Rd,p obab wi h ein o cemen a io. Fo low ρ- alues, he sys em esis ance is go e ned by he ein o cemen con ibu ion while he conc e e con ibu ion becomes mo e impo an wi h inc easing ρ. I appea s ha PFM p o ides a good app oxima ion in he case o a se ies sys em. The me hod is ocused on c i ical membe s and hus is well sui ed o he analysis o se ies sys ems. No e ha his conclusion is alid only o well-calib a ed alues o he pa ial ac o s o esis ances ( ailu e modes) unde conside a ion. Fig. 1: Va iabili y o Rd,PFM / Rd,p obab and Rd,ECoV / Rd,p obab wi h ρ (Case 1). R d,semi-p ob / R d,p obab ρ 0.95 1 1.05 1.1 3% 4% 5% 6% 7% 8% 1.15 1.2 R d,PFM /R d,p obab ρ m = cm / ym ρ k = ck / yk R d,ECoV /R d,p obab ρ d = ( ck /γ C )/ ( yk /γ S ) SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 83 Fig. 2: Va iabili y o s a is ical cha ac e is ics es ima ed by ECoV and p obabilis ic app oach wi h ρ ( om op o bo om – mean alues, coe icien s o a ia ion, and coe icien s o skewness). In con as , ECoV may o e es ima e up o 18% o ρ- alues anging om ρd o ρm. App oxima ely o ρ > ρd, he conc e e con ibu ion becomes decisi e o sys em esis ance while ECoV iden i ies a domina ing s eel con ibu ion by: • bo h analyses based on he mean and cha ac e is ic alues o ρd < ρ < ρk, • by analysis based on he mean alues o ρk < ρ < ρm. To u he in es iga e hese de iciencies, Fig. 2 shows a iabili y o mean alues, coe icien s o a ia ion, and coe icien s o skewness o sys em esis ance es ima ed by ECoV and by he p obabilis ic app oach. The unsa e e o o 18% obse ed o Rd,ECoV(ρk) is a ibu ed o he ollowing: 1. Key is ailu e in iden i ying a ype o dis ibu ion o sys em esis ance – igno ing he bimodal cha ac e o he dis ibu ion and wi hou ega d o an ac ual coe icien o skewness (Fig. 2 – posi i e ωECoV leads o a highe design esis ance in compa ison o nega i e ωp obab). De ailed analysis indica es ha his aspec leads o an e o o abou 13%. 2. O he wo less impo an e ec s con ibu ing o he unsa e e o a e a small o e es ima ion o he mean alue (3%) and unde es ima ion o coe icien o a ia ion (2%). In e es ing o no e is ha he e o nea ly anishes o Rd,ECoV(ρm). Failu e o iden i y a ype o dis ibu ion and o e es ima ed mean would lead o a signi ican o e es ima ion, 16% + 10%, bu hey a e ou weighed by a ma kedly o e es ima ed coe icien o a ia ion (VR,ECoV(ρm) = 21.3% while VR,p obab(ρm) = 13.4%). Wi h inc easing ein o cemen a io, he e o s dec ease and anish as no sys em beha iou occu s wi h he conc e e con ibu ion en i ely domina ing. 5. No mal Componen Resis ances (Case 2) Case 2 in es iga es he same se ies sys em as in Case 1, bu he conc e e and yield con ibu ions a e now desc ibed by no mal dis ibu ion wi h he ollowing adjus ed cha ac e is ics: • V c’ = 14.4%, and c0.05’ = 22.2 MPa, • V y’ = 8.1%, and y0.05’ = 394 MPa. while he mean alues, cm’ and ym’ emain unchanged. Simila ly as in Equa ions (8) and (9), he coe icien s o a ia ion a e de e mined o co espond o he pa ial ac o s: cd’ = cm’ (1-αR β V c’ ) = ck’ / 1.35, (10) yd’ = ym’ (1-αR β V y’ ) = yk’ / 1.15. (11) Fig. 3 displays a iabili y o Rd,PFM / Rd,p obab and Rd,ECoV / Rd,p obab wi h ein o cemen a io. Va ia ion o he a ios is simila o ha obse ed in Case 1. The ECoV e o eaches up o 20% o Rd,ECoV(ρk) and con e ging o 8% e en o high ein o cemen a ios. Despi e he la e is he case wi h one domina ing componen (conc e e), ECoV assumes a logno mal dis ibu ion which is inadequa e o μ R (MPa) ρ 15 18 21 24 3% 4% 5% 6% 7% 8% 27 30 ρ d ρ m ρ k ECoV p obabilis ic V R 0% 5% 10% 15% 3% 4% 5% 6% 7% 8% 20% 25% ρ ρ d ρ m ρ k ECoV p obabilis ic ω R -0.5 -0.25 0 0.25 3% 4% 5% 6% 7% 8% 0.5 0.75 ρ ECoV p obabilis ic ρ d ρ m ρ k SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 84 no mally dis ibu ed conc e e s eng h. Fo low a ios, his e o is also p esen , bu i is o small magni ude due o a low coe icien o a ia ion o esis ance ela ed o s eel yielding. The unsa e app oxima ion by ECoV is now p ima ily a ibu ed o ailu e in iden i ying a ype o dis ibu ion o sys em esis ance. No ak L. and No ak D. [8] adjus ed ECoV o he si ua ion when a no mal dis ibu ion is assumed o esis ances. Fig. 3 shows ha his adjus men co ec s he ECoV es ima es o low and high ρ- alues whe e esis ances o indi idual componen s a e domina ing. Howe e as in Sec ion 4, he unsa e e o o abou 20% is ound o a ios close o ρk. Fig. 3: Va iabili y o Rd,PFM / Rd,p obab and Rd,ECoV / Rd,p obab wi h ρ (Case 2). 6. Discussion The p esen ed limi ed analysis o he se ies sys ems wi h wo ailu e modes indica es some di ec ions o u he esea ch: 1. Mos conc e e s uc u al sys ems a e deemed o ha e p ope ies close o pa allel sys ems as hey a e o en inde e mina e, p o iding o mul iple load pa hs. P elimina y esul s o pa allel sys ems, pa ly p esen ed in [5], indica e ha ECoV pe o ms e y well while PFM ends o be a he conse a i e o pa allel sys ems when αR = 0.8 is conside ed o bo h conc e e and s eel. 2. Posi i e co ela ions be ween ailu e modes a e expec ed o educe he ECoV e o o bo h ypes o sys ems. In con as , he e o s may ampli y wi h an inc easing numbe o ailu e modes o simila impo ance. These coun e ac ing e ec s should be nume ically in es iga ed in he u u e. 3. The e o in cases wi h non-logno mal sys em esis ance may be signi ican . In his con ex , si ua ions wi h impo an a iabili y o geome ical p ope ies (pa icula ly o small-size membe s) o wi h non- logno mal ma e ial p ope ies (possibly o UHPC) should be analyzed. 4. La in Hype cube Sampling (LHS) wi h a low numbe o simula ions—also included in he d a ib MC 2020— is o en applied o e i y he pe o mance o simpli ied semi-p obabilis ic me hods. S a is ical unce ain y in LHS es ima es o cases wi h mul iple ailu e modes emains o be in es iga ed and quan i ied. 5. Fig. 2 clea ly shows ha ECoV migh conside ably o e es ima e coe icien o a ia ion o sys em esis ance. The ad anced a iance educ ion app oaches such as Taylo se ies expansion o Eigen ECoV [8] may p o ide signi ican imp o emen s, bu equi e mo e complex in o ma ion abou s ochas ic models and ailu e egions. Bene i s and cos s ela ed o he applica ions o hese ad anced me hods should be u he explo ed. 7. Concluding Rema ks P e ious pilo in es iga ions e ealed ha he Me hod o Es ima ion o Coe icien o Va ia ion (ECoV) migh o e es ima e he design esis ance o some se ies sys ems. A de ailed analysis o wo s udy cases using ECoV and he pa ial ac o me hod (PFM) p esen ed he e demons a es ha o he se ies sys ems unde conside a ion: • PFM p o ides good app oxima ions. The me hod is ocused on c i ical membe s and hus is well sui ed o he analysis o se ies sys ems. Howe e , his conclusion is alid only o well-calib a ed alues o he pa ial ac o s o esis ances ( ailu e modes) unde conside a ion. • ECoV may o e es ima e up o abou 20% in si ua ions whe e NLFEA based on mean and/ o cha ac e is ic alues ail o iden i y a domina ing ailu e mode. • The maximum obse ed e o is a ibu ed mainly o ailu e in iden i ying a ype o dis ibu ion o sys em esis ance; he o he wo, less impo an a e a small o e es ima ion o he mean alue and unde es ima ion o coe icien o a ia ion. The p esen ed limi ed analysis indica es a numbe o di ec ions o u he esea ch, including he analysis o : • pa allel and mixed se ies-pa allel sys ems, • cases wi h s ongly non-logno mal esis ance, • e ec s o many componen s o he sys em and co ela ions be ween componen esis ances, • pe o mance o ad anced me hods o eliabili y e i ica ion o RC s uc u es. In gene al, ecommenda ions on how o iden i y si ua ions when a signi ican e o can be expec ed should be p o ided o p ac ical applica ions o simpli ied semi- p obabilis ic me hods. R d,semi-p ob / R d,p obab ρ 0.95 1 1.05 1.1 3% 4% 5% 6% 7% 8% 1.15 1.2 R d,PFM /R d,p obab R d,ECoV /R d,p obab R d,ECoV /R d,p obab adjus ed o no mal dis . ρ m ρ k ρ d SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 21 | NUMBER: 2 | 2021 | DECEMBER © 2021 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 85 Acknowledgemen s This s udy has been suppo ed by he Czech Science Founda ion unde G an 20-01781S. Re e ences [1] ALLAIX, D.L., V.I. CARBONE and G. MANCINI. Global sa e y o ma o non-linea analysis o ein o ced conc e e s uc u es. S uc . Conc . 2013, Vol. 14, N . 1, pp. 29-42. ISSN 1751-7648. DOI: 10.1002/suco.201200017. [2] CERVENKA, V. Global Sa e y Fo ma o Nonlinea Calcula ion o Rein o ced Conc e e. Be on- und S ahlbe onbau. 2008, Vol. 103, N . 2008, pp. 37-42. DOI: 10.1002/bes .200810117. [3] CERVENKA, V. Reliabili y-based non-linea analysis acco ding o ib Model Code 2010. S uc . Conc . 2013, Vol. 14, N . 1, pp. 19-28. ISSN 1751- 7648. DOI: 10.1002/suco.201200022. [4] CERVENKA, J., V. CERVENKA, M. SYKORA and J. MLCOCH. E alua ion o Sa e y Fo ma s o S uc u al Assessmen Based on Nonlinea Analysis. In P oc. EURO-C 2018. London: CRC P ess, 2018, pp. 669-678. ISBN 9781351726764. [5] SYKORA, M., J. CERVENKA, V. CERVENKA, J. MLCOCH, D. NOVAK and L. NOVAK. Pilo Compa ison o Sa e y Fo ma s o Reliabili y Assessmen o RC S uc u es. In P oc. ib Symp. 2019. Lausanne: In e na ional Fede a ion o S uc u al Conc e e, 2019, ISBN 9782940643004. [6] HOLICKY, M. In oduc ion o P obabili y and S a is ics o Enginee s. Be lin: Sp inge -Ve lag, 2013. 181 pp. ISBN 978-3-642-38299-4. DOI: 10.1007/978-3-642-38300-7. [7] EUROPEAN CONCRETE PLATFORM. Eu ocode 2 Commen a y. 2008. [8] NOVAK, L. and D. NOVAK. Es ima ion o coe icien o a ia ion o s uc u al analysis: The co ela ion in e al app oach. S uc .Sa . 2021, Vol. 92, pp. 102101. ISSN 0167-4730. DOI: h ps://doi.o g/10.1016/j.s usa e.2021.102101. Abou Au ho s Mi osla SÝKORA was bo n in České Budějo ice, Czech Republic. He was appoin ed Associa e P o esso a Facul y o Ci il Enginee ing, CTU in P ague in 2015. His esea ch in e es s include he basis o s uc u al design and o assessmen o exis ing s uc u es, s uc u al eliabili y, p obabilis ic op imisa ion, load modelling, isk assessmen o echnical sys ems, and applica ions o p obabilis ic me hods in s uc u al design. Jana MARKOVÁ was bo n in P ague, Czech Republic. She was appoin ed Associa e P o esso a Facul y o Ci il Enginee ing, CTU in P ague in 2007. He esea ch in e es s include basis o s uc u al design and assessmen o exis ing s uc u es, s uc u al eliabili y, load modelling wi h special ocus on clima ic ac ion e ec s, in es iga ions o clima e change e ec s on s uc u al eliabili y, and applica ions o p obabilis ic me hods in s uc u al design. Vi ali NADOLSKI was bo n in Minsk, Bela us. He ecei ed his Ph.D. deg ee om he B es S a e Technical Uni e si y in 2015. His esea ch in e es s include eliabili y analyses, FEA-based design, and esis ance models o s eel s uc u es.