Limit states of structures and global sensitivity analysis based on Cramér-von Mises distance

Abs ac — This a icle p esen s a s ochas ic compu a ional model o he analysis o he eliabili y o a d awn s eel ba . The whole dis ibu ion o he limi s a e unc ion is s udied using global sensi i i y analysis based on C amé - on Mises dis ance. The algo i hm o es ima ing he sensi i i y indices is based on one loop o he La in Hype cube Sampling me hod in combina ion wi h nume ical in eg a ion. The algo i hm is e ec i e due o he app oxima ion o esis ance using a h ee- pa ame e logno mal dis ibu ion. Goodness-o - i es s and o he compa a i e s udies demons a e he signi ican accu acy and sui abili y o he h ee-pa ame e logno mal dis ibu ion, which p o ides be e esul s and as e esponse han sampling-based me hods. Global sensi i i y analysis is e alua ed o wo load cases wi h p o en dominan e ec o he long- e m a ia ion load ac ion, which is in oduced using Gumbel p obabili y densi y unc ion. The C amé - on Mises indices a e discussed in he con ex o o he ypes o p obabili y-o ien ed sensi i i y indices whose pe o mance has been s udied ea lie . Keywo ds— Global sensi i i y analysis, s eel, p obabili y, ailu e, eliabili y, andom sampling. I. INTRODUCTION eliabili y is desc ibed as he abili y o a sys em o componen o unc ion unde s a ed condi ions o a speci ied pe iod o ime [1, 2]. S uc u al eliabili y is assessed using me hods o p obabilis ic analysis [3]. S uc u al sensi i i y analysis is a sui able complemen o p obabilis ic eliabili y analysis [4]. The basic measu e o eliabili y is he p obabili y ha ailu e o a load-bea ing s uc u e does no occu [5]. The mos se ious ailu e is loss o load-ca ying capaci y o a componen o membe wi hin a s uc u e o o he s uc u e i sel , see o e.g. [6]. Failu e o a s uc u e occu s when he ma e ial in he s uc u e is s essed o i s s eng h limi [7]. Al hough s esses canno be di ec ly in es iga ed expe imen ally, limi s a es can be in es iga ed by measu ing pe manen de o ma ions o obse ing ac u es [8]. Load-bea ing s uc u es mus be designed wi h such geome ic and ma e ial cha ac e is ics ha he p obabili y o ailu e is e y low. Fo eliable design, p obabili y o 7.2E-5 o less is commonly equi ed [5]. A speci ic ype o ailu e o slende s eel s uc u es is loss o s abili y, which causes he s uc u e o collapse be o e eaching i s ma e ial s eng h limi [9]. In s ochas ic sys ems, s abili y o en means insensi i i y o low sensi i i y o hei ou pu cha ac e is ics o he shapes o some inpu dis ibu ions [10]. In ope a ions esea ch, sensi i i y analysis is de eloped as a me hod o c i ical assessmen o decisional a iables, which ha e in e ela ions among hem, and is capable o iden i ying hose sensi i e a iables ha ha e an in luence on he inal desi ed esul [11-12]. One o he p incipal p oblems in sys em eliabili y s udies is he p oblem o hei sensi i i y o he shape o inpu dis ibu ions [13-15]. The desi ed esul may be he alida ion o he a ibu es o s a ic [16] o dynamic [17] models ha equi e simul aneous op imiza ion, bu also o he pe o mance cha ac e is ics associa ed wi h he s uc u e, see o e.g. [18-19]. Assessmen o decisional a iables can be based on MCDM [20] and compu e expe imen s [21]. The basic calib a ion ools o eliabili y acco ding o s anda ds o s uc u al design a e he pa ial sa e y ac o s o load and esis ance [22-25]. The e i ica ion o eliabili y o s anda d design is commonly pe o med using p obabilis ic me hods [26, 27], bu only de e minis ic ypes o sensi i i y s udies, see o e.g. [28, 29]. The ask o sensi i i y analysis in Limi s a es o s uc u es and global sensi i i y analysis based on C amé - on Mises dis ance Z. Kala, Depa men o S uc u al Mechanics, Facul y o Ci il Enginee ing, B no Uni e si y o Technology Ve eří S . 95, B no, ZIP 602 00 Czech Republic Recei ed: May 5, 2020. Re ised: June 27, 2020. Accep ed: July 3, 2020. Published: July 7, 2020. R INTERNATIONAL JOURNAL OF MECHANICS DOI: 10.46300/9104.2020.14.14 Volume 14, 2020 ISSN: 1998-4448 107 s ochas ic models o ien ed o s uc u al eliabili y is o quan i y he e ec s o inpu a iables on he ou pu cha ac e is ics, he choice o which is speci ic o he gi en ype o s uc u e and compu a ional model, see e.g. [30-35]. Acco ding o con en ional de ini ion, eliabili y sensi i i y analysis ocuses on measu ing he e ec s o unce ain y o dis ibu ion pa ame e s o inpu a iables on ailu e p obabili y [36]. Sensi i i y analysis subo dina ed o con as s [37] appea s o be an app op ia e eliabili y-o ien ed me hod, which can be used o analyse bo h he p obabili y o ailu e [38-39] and design quan iles [40-42]. Ano he me hod discussed is global sensi i i y analysis based on C amé - on Mises dis ance [43], which is ocused on he dis ibu ion unc ion o he esponse. O he ypes o sensi i i y analysis based on measu ing he absolu e di e ence be ween he uncondi ional and condi ional ailu e p obabili y we e published in [44-45]. Bo gono o momen independen impo ance measu e [46] was no de eloped di ec ly o he analysis o ailu e p obabili y bu can be men ioned as one o he ew me hods ha pe mi s he analysis o s a is ically dependen inpu andom a iables. The analysis and compa ison o di e en echniques o sensi i i y analysis and a discussion on hei s eng hs and weaknesses has been published in [47]. I an analy ical solu ion is no a ailable, he ailu e p obabili y can be es ima ed using me hods such as Mon e Ca lo, Impo ance Sampling [48], Adap i e Sampling [49], Response Su ace [50], a i icial neu al ne wo ks [51] o polynomial chaos expansions [52]. Un o una ely, a plan ailo ed o es ima e a pa icula ype o sensi i i y index, o se he eo , may be subop imal o o he indices. In his pape , he e ec s o inpu andom a iables on he dis ibu ion unc ion o he limi s a e a e s udied using C amé - on Mises indices. One o he objec i es o his pape is o de elop a s ochas ic compu a ional model o a load-bea ing elemen ha will acili a e he e icien e alua ion o global sensi i i y analysis and which will be used in u he esea ch. II. STOCHASTIC MODEL AND RELIABILITY ANALYSIS The aim o he s udy is o analyse he heo e ical eliabili y o a s eel s uc u e, which is a he design s age. This is p ima ily ensu ed by design eliabili y condi ions o s anda d EN 1990 [5]. A. Load Ac ion Le a d awn ba be subjec ed o a combina ion o pe manen load ac ion G and long- e m leading single a iable load ac ion Q. Pe manen load ac ion G can be conside ed wi h mean alue µ G, which is equal o he cha ac e is ic alue Gk [53]. The andom de ia ions o he pe manen load ac ion om he mean alue should be symme ical, which co esponds well wi h he Gauss p obabili y densi y unc ion, see Fig. 1. The s anda d de ia ion σ G is selec ed unde he assump ion ha app oxima ely 50% o andom ealiza ions o G a e ound in he in e al ± σ G. Fig. 1 Densi y p obabili y unc ions o pe manen load ac ions The leading a iable load ac ion is in oduced wi h Gumbel p obabili y densi y unc ion [54], see Fig. 2, which is he cha ac e is ic unc ion o ex eme alue dis ibu ions [55, 56]. The cumula i e dis ibu ion unc ion o Gumbel dis ibu ion is ( )             − −−= β α x xF expexp , (1) o −∞ < x < ∞, −∞ < α < ∞ and β > 0. Pa ame e s α and β can be exp essed om βγαµ += Q , (2) 6 πβ σ = Q , (3) whe e γ=0.5772156649 is he Eule –Masche oni cons an . The mean alue µ Q and s anda d de ia ion σ Q a e in oduced as unc ions o he cha ac e is ic alue Qk using he equa ions µ Q=0.6 Qk, σ Q=0.21 Qk, which we e used in [53]. Fig. 2 Densi y p obabili y unc ions o a iable load ac ions I can be no ed ha Gauss and Gumbel p obabili y densi y unc ions a e dis ibu ions ecommended in s anda d EN1990 [5] and can be used in he s udies p esen ed he e, see Table I. INTERNATIONAL JOURNAL OF MECHANICS DOI: 10.46300/9104.2020.14.14 Volume 14, 2020 ISSN: 1998-4448 108 TABLE I INPUT RANDOM VARIABLES OF LOAD ACTION Cha ac e is ic Pd Mean alue S . de ia ion Pe manen load G Va iable load ac ion Q Gauss Gumbel-max G k 0.6 ⋅ Qk 0.1⋅G k 0.21⋅Qk Howe e , o he ypes o unc ions may also be used in jus i ied cases o speci ic load cases. A de ailed desc ip ion o hese and o he less equen ly used heo e ical models o andom a iables can be ound in In e na ional S anda ds ISO 13822, ISO 12491, ISO 3534-1 and specialized li e a u e [57]. B. Resis ance The d awn ba is made om a s eel shee o leng h L, wid h b and hickness . Wid h and hickness a e he dimensions o he ec angula c oss-sec ion. The esis ance R is exp essed as he p oduc o he yield s eng h y and c oss-sec ional a ea A o he ec angula c oss-sec ion. A R y ⋅= , (4) whe e b A ⋅= . (5) The yield s eng h, which is gene ally a andom a iable, is an impo an ma e ial cha ac e is ic o s eel. In he pas , nume ous expe imen al es s on he ma e ial and geome ic cha ac e is ics o s uc u al s eel ha e been pe o med [58], which p o ide impo an backg ound in o ma ion o he analysis o esis ance. The inpu andom a iables a e lis ed in Table II. TABLE II INPUT RANDOM VARIABLES OF RESISTANCE PARAMETERS Cha ac e is ic Pd Mean alue S . de ia ion Yield s eng h y Specimen hickness Specimen wid h b Gauss Gauss Gauss 395.68 MPa 16 mm 80 mm 25.13 MPa 0.734 mm 0.8 mm Inpu a iable A has an asymme ic chi squa e p obabili y densi y unc ion. Resis ance R is he p oduc o he yield s eng h and he c oss-sec ional a ea gi en by Eq. (5). I we mul iply h ee andom a iables y⋅ ⋅b wi h Gauss p obabili y densi y unc ions, he compu ed esis ance R does no ha e a Gauss p obabili y densi y unc ion (has an asymme ic dis ibu ion) and i is necessa y o seek he mos sui able dis ibu ion o he app oxima ion o he dis ibu ion o R. I can be no ed ha he p obabili y densi y dis ibu ion o R can be analysed by he nume ical (bu no analy ical) in eg a ion o a iple in eg al o by app oxima ing he Gauss p obabili y densi y unc ion om Table II using such unc ions ha can be in eg a ed analy ically. This app oach would ha e o be used i none o he known ypes o p obabili y densi y unc ions p o ed o be sui able o app oxima ing he p obabili y densi y unc ion o R. A sui able app oxima ion o he p obabili y densi y unc ion o andom a iable R can be selec ed om nume ous ypes o p obabili y densi y unc ions among which ce ain ypes can be i ed mo e closely o he obse ed equency o he esis ance han o he s, depending on he cha ac e is ics o he phenomenon and o he dis ibu ion. One o he possibili ies o app oxima ing R in he case s udy analysed he ein is by using he h ee-pa ame e logno mal dis ibu ion. The so wa e S a el [59] e e s o he h ee-pa ame e logno mal dis ibu ion, which will be used below, as he shi ed logno mal p obabili y densi y unc ion. ( ) ( ) ( )( )       ⋅ −− − ⋅⋅− = 2 2 0 0 2 ln exp 2 1 u u u S mxx Sxx x π , (6) o x0 < x < ∞, whe e xx x σµ ⋅+= 00 , (7)        += 2 0 1 1ln S u , (8) 2 ln 2 0 u xu S m −⋅= σ , (9) whe e x x xxxx a a azaz 1 2 22 3 32 32 0 − ⋅ ++−−− = , (10) whe e ax is s anda dized skewness and 4 2+= xxx aaz . (11) C. P obabili y Analysis o Resis ance The selec ion o he sui able p obabili y densi y unc ion o esis ance was e i ied nume ically by compa ing he esul s o wo p obabilis ic s udies. I was es ed whe he he shi ed logno mal p obabili y densi y unc ion gi es a close i and leads o good p edic ions. The i s p obabili y IP R is calcula ed by nume ical in eg a ion using he shi ed logno mal p obabili y densi y unc ion. The second p obabili y IIP R is calcula ed by nume ical in eg a ion o he unc ion R= y ⋅ ⋅ b in h ee- dimensional space. The alue IIP R can be desc ibed as a e y good app oxima ion o he “ eal” p obabili y ha could heo e ically be achie ed i he in eg a ion s ep app oaches ze o. ( ) i R RPP <= , (12) The sui abili y o he shi ed logno mal p obabili y densi y unc ion is e i ied by compa ing he alues o IP R and IIP R. The inaccu acies o he es ima es o bo h IP R and IIP R a e INTERNATIONAL JOURNAL OF MECHANICS DOI: 10.46300/9104.2020.14.14 Volume 14, 2020 ISSN: 1998-4448 109 negligible because nume ical in eg a ion was pe o med wi h 7000000 (IP R) and 70003 s eps, see he a iable max in he algo i hm below. Th oughou he a icle, nume ical in eg a ion is pe o med on he in e al mean alue plus minus en s anda d de ia ions o he in eg a ed a iable. The in oduc ion o wide in e als has a negligible e ec on inc easing he accu acy o he esul s o he nume ical in eg a ions. Th ee nes ed loops a e used in he algo i hm o he nume ical in eg a ion o he compu a ion o IIP R. The basic o m o he nume ical in eg a ion algo i hm can clea ly be w i en in he p og amming language Pascal as: unc ion No mR2(x,m,S:ex ended):ex ended; begin No mR2:=exp(-sq (x-m)/(2*sq (S)))/(sq (2*pi)*S); end; {Th ee nes ed loops nume ical in eg a ion} Func ion P (R:Ex ended):Ex ended; a i1,i2,i3,max :in ege ; a1,a2,a3,b1,b2,b3 :ex ended; P R,Sm10,R04,R01,R02,R03 :ex ended; m1,m2,m3,S1,S2,S3 :ex ended; begin max:=7000; // s eps o nume ical in eg a ion m1:=395.68;m2:=16;m3:=80; S1:=25.13;S2:=0.734;S3:=0.8; R:=200E3; // R is alue o esis ance Sm10:=10; R01:=m1-Sm10*S1; R02:=m2-Sm10*S2; R03:=m3-Sm10*S3; b1:=R01; P R:=0; // P R is p obabili y o R o i1:=1 o max do begin a1:=b1;b1:=R01+(m1+Sm10*S1-R01)*i1/max; b2:=R02; o i2:=1 o max do begin a2:=b2;b2:=R02+(m2+Sm10*S2-R02)*i2/max; R04:=(b1-a1)*(b2-a2); b3:=R03; o i3:=1 o max do begin a3:=b3;b3:=R03+(m3+Sm10*S3-R03)*i3/max; i (a1+b1)*(a2+b2)*(a3+b3)/8<R hen P R:=P R+R04*(b3-a3)*No mR2((a1+b1)/2,m1,S1)* No mR2((a2+b2)/2,m2,S2)*No mR2((a3+b3)/2,m3,S3); end; end; end; P :=P R; // P is p obabili y o R end; The compu a ional complexi y o he algo i hm is 1012 s eps o mainly a i hme ic ope a ions. The algo i hm uns o 70 minu es on a single co e CPU In el Co e i7-3740QM p ocesso . I can be no ed ha a simila algo i hm is applied in he DOP oC me hod [60], which, howe e , has inpu andom a iables de ined by his og ams. The assump ion o using h ee nes ed loops o es ima e IIP R is demanding on CPU ime e en hough i is a ela i ely simple algo i hm. The es ima ion o IIP R is mo e nume ically demanding he highe he alue o IIP R. The e o e, o IIP R>0.5 i is possible o speed up he algo i hm by modi ying he condi ion in o he o m “(a1+b1)*(a2+b2)*(a3+b3)/8>=R“ and “P :=1-P R;“, which is a key op imiza ion. Addi ional op imiza ions we e also pe o med. The compu a ion o IP R using he shi ed logno mal p obabili y densi y unc ion is no so nume ically demanding and was he e o e pe o med using se en million s eps. The compu a ional complexi y o he calcula ion o IP R is app oxima ely one hund ed housand imes lowe han he calcula ion o IIP R. Table III con ains a compa ison o IIP R alues wi h IP R alues. F om a nume ical poin o iew, i is appa en ha he es ima e o IP R is su icien ly accu a e because he di e ences be ween he es ima es o IP R and IIP R a e minimal, wi h he excep ion o e y small p obabili ies. Con o mi y in exponen s wi h he powe o E-19 is ully su icien in e ms o enginee ing p obabili y o ailu es. Possible limi a ions pe ain o p obabili ies wi h exponen s lowe han E-19, which, howe e , is i ele an in enginee ing applica ions. TABLE III COMPARISON OF PROBABILITY COMPUTATIONS Resis ance i IP R Shi ed logno mal pd IIP R Nume ical in eg a ion using h ee nes ed loops 0 100 kN 200 kN 300 kN 400 kN 500 kN 600 kN 700 kN 2.6215E-65 2.7419E-37 1.3556E-19 5.8430E-09 2.7029E-03 4.4285E-01 9.8810E-01 9.99996E-01 0 3.1258E-47 1.1362E-19 5.2958E-09 2.6880E-03 4.4288E-01 9.8812E-01 9.99996E-01 The esis ance mus be posi i e. This is one o he a gumen s ha some imes calls in o ques ion he sui abili y o app oxima ing esis ance using he shi ed logno mal p obabili y densi y unc ion in he e en ha i is also de ined o nega i e alues. The shi ed logno mal p obabili y densi y unc ion used in his pape is de ined on he in e al (-579.377 kN; ∞). I is e iden om Table III ha he p obabili y ha he esis ance has a nega i e alue is IP R=2.6215E-65. This is a negligible alue. I we in oduce a immed shi ed logno mal p obabili y densi y unc ion, he alue o he ob ained p obabili y is p ac ically he same. This implies ha imming is no necessa y. The shi ed logno mal p obabili y densi y unc ion can be used as i is. Table IV con ains a compa ison o IIP R wi h p obabili y IIIP R ob ained by app oxima ion using Gauss p obabili y densi y unc ion, which is no conside ed app op ia e. I is appa en ha he es ima e IP R (Shi ed logno mal p obabili y densi y unc ion) is signi ican ly mo e accu a e in compa ison INTERNATIONAL JOURNAL OF MECHANICS DOI: 10.46300/9104.2020.14.14 Volume 14, 2020 ISSN: 1998-4448 110 wi h IIIP R, see Table IV. Es ima es o IIIP R ob ained by app oxima ion using Gauss p obabili y densi y unc ion ha e disc epancies om he a ge IIP R o small p obabili ies. TABLE IV COMPARISON OF PROBABILITY COMPUTATIONS Resis ance i IIIP R Gauss pd IIP R Nume ical in eg a ion using h ee nes ed loops 0 100 kN 200 kN 300 kN 400 kN 500 kN 600 kN 700 kN 5.4609E-37 1.5929E-24 9.6066E-15 1.2498E-07 3.9107E-03 4.3580E-01 9.9027E-01 9.999999E-01 0 3.1258E-47 1.1362E-19 5.2958E-09 2.6880E-03 4.4288E-01 9.8812E-01 9.99996E-01 The compa a i e s udies con i med ha he shi ed logno mal p obabili y densi y unc ion is a sui able app oxima ion o esis ance compu ed om Eq. (4). The ollowing chap e alida es his conclusion using goodness-o - i es s. D. Sampling Based and S a is ical Analysis o Resis ance The goodness-o - i es can be used o es i sample da a i a dis ibu ion om a andom obse a ion. The andom obse a ions we e ob ained using he La in Hype cube Sampling Me hod (LHS) [61, 62]. One million s eps o he LHS me hod we e used o simula e he andom ealiza ions o R. The use o he LHS me hod o such a la ge numbe o samples has p o en app op ia e since he esul s ob ained using he LHS me hod a e always be e han he esul s ob ained using he simple Mon e Ca lo me hod, including he la ge numbe s o uns, which is use ul in global sensi i i y analyses o he limi s a e unc ion dis ibu ion [38]. Fig. 3 App oxima ion o he his og am o esis ance The esul s o he s a is ical analysis o esis ance R ob ained using he LHS me hod a e shown in Table V. Th ee a ian s o he app oxima ion o he his og am o esis ance compu ed using he inpu andom a iables lis ed in Table II a e shown in Fig. 3. A i hme ic mean µ R, s anda d de ia ion σ R and s anda d skewness aR a e impo an esul s o he s a is ical analysis, which a e also applicable as pa ame e s o he es ed shi ed logno mal p obabili y densi y unc ion. The alues o µ R, σ R and aR can be compu ed analy ically om he unc ion Y 321 XXXY ⋅⋅= , (13) o h ee andom a iables X1, X2, X3 wi h Gauss p obabili y densi y unc ions. Each andom a iable Xi is cha ac e ized by i s a i hme ic mean µ i and s anda d de ia ion σ i. A i hme ic mean µ Y, s anda d de ia ion σ Y and s anda d skewness aY o andom a iable Y a e hen exp essed by he ollowing equa ions: 321 µµµµ ⋅⋅= Y , (14) ( ) γσσγσµγσσµµσ ⋅⋅+⋅⋅+⋅+⋅⋅= 2 2 2 1 2 1 2 2 2 2 2 3 2 2 2 1Y , (15) whe e 2 3 2 3 σµγ += , (16) ( ) .4 6 2 3 2 2 2 1 2 3 2 2 2 1 2 3 2 2 2 1 2 3 2 2 2 1 3 σσσ µσσσµσσσµ σ µ ⋅⋅⋅+ ⋅⋅+⋅⋅+⋅⋅⋅⋅= Y Y Y a , (17) The s a is ical cha ac e is ics o esis ance R Eq. (4) compu ed in his s udy using Eq. (14), (15), (17) and Table II a e µ R=506.47 kN, σ R=40.031 kN and aR= 0.11065. The compa ison o µ R, σ R and aR wi h he esul s in Table V shows pe ec ag eemen be ween he analy ical solu ion and he s a is ical analysis. TABLE V STATISTICAL CHARACTERISTICS OF HISTOGRAM IN FIG. 3 Cha ac e is ic Value Valid obse a ions Minimum Maximum Range Median A i hme ic mean Geome ic mean Mean squa e Va iance S and. de ia ion Coe . o a ia ion Thi d momen S and. skewness Fou h momen S and. ku osis Va iance o mean Va . o a iance Va . o 3. momen Va . o 4. momen 1000000 323.62 710.63 387.01 505.73 506.47 504.89 1602.4 1602.4 40.030 0.79037E-01 7041.1 0.10977 0.77691E+07 3.0257 0.16023E-02 0.60313E+08 26005. 0.68249E+09 INTERNATIONAL JOURNAL OF MECHANICS DOI: 10.46300/9104.2020.14.14 Volume 14, 2020 ISSN: 1998-4448 111 E. Goodness-o - i Tes s o Resis ance Goodness-o - i es s a e s a is ical ools ha can eliably ejec app oxima ions ha a e no pe ec ly adap ed o da a ob ained using he LHS me hod. Measu es o goodness-o - i ypically summa ize he disc epancy be ween obse ed alues and he alues expec ed unde he model in ques ion. Goodness-o - i es s may ejec unsui able ypes o app oxima ions wi h a selec ed p obabili y, bu hey canno con i m he app op ia e ypes o p obabili y densi y unc ions. Thus, he aim o he s udy can only be o ejec inapp op ia e ypes o p obabili y densi y unc ions o he app oxima ion o he esis ance. The es ed models in ques ion a e shi ed logno mal, Gauss, Logno mal, He mi e, Gamma and Be a p obabili y densi y unc ions. Chi-squa e, Kolmogo o - Smi no and Ande son-Da ling goodness-o - i es s a e used, see Table VI. Tes ing o he shi ed logno mal p obabili y densi y unc ion is depic ed in Fig. 4, Fig. 5 and Fig. 6. The es s we e pe o med using he p og am S a el 3.10. Fig. 4 Chi-squa e goodness-o - i es [59] o shi ed Logno mal pd Fig. 5 Kolmogo o -Smi no goodness-o - i es [59] o shi ed Logno mal pd Based on he esul s o h ee goodness-o - i es s, i can be concluded ha he shi ed logno mal p obabili y densi y unc ion canno be ejec ed o he app oxima ion o he p obabili y densi y unc ion o R. Fig. 6 Ande son-Da ling goodness-o - i es [59] o shi ed Logno mal pd I is e iden om he esul s in Table VI ha he conclusions o Chi-squa e and Kolmogo o -Smi no es s a e he same and a e consis en wi h he esul s o he s udies in he p e ious chap e s. The Ande son-Da ling es did no ejec he hypo hesis in any case, e en hough, o example, in Table IV i was clea ly shown ha he choice o Gauss p obabili y densi y unc ion is no sui able o he app oxima ion o he p obabili y densi y unc ion o R. In addi ion o he shi ed logno mal p obabili y densi y unc ion, ou -pa ame e He mi e p obabili y densi y unc ion could also be conside ed. TABLE VI GOODNESS-OF-FIT TEST OF HISTOGRAM IN FIG. 3 P obabili y densi y unc ion Chi-squa e es Kolmogo o - Smi no es Ande son- Da ling es Shi ed Logno mal Gauss Logno mal He mi e (S a el) Gamma Be a No ejec ed Rejec ed Rejec ed No ejec ed Rejec ed Rejec ed No ejec ed Rejec ed Rejec ed No ejec ed Rejec ed Rejec ed No ejec ed No ejec ed No ejec ed No ejec ed No ejec ed No ejec ed III. PROBABILITY ASSESSMENT OF RELIABILITY Reliabili y can be unde s ood as he abili y o a s uc u e o a s uc u al membe o ul il s a ed equi emen s o which i has been designed. Reliabili y is gene ally exp essed in e ms o p obabili y. The ul ima e limi s a es a e de ined as s a es ha a e associa ed wi h collapse o wi h simila o ms o s uc u al ailu e. Loads highe han esis ance can be unde s ood as andom ailu e, which occu s in dependence on andom ma e ial and geome ic cha ac e is ics o s uc u al membe s and load ac ions. The p obabili y o ailu e can be w i en as: ( ) RQGPP >+= , (18) whe e G, Q and R a e andom a iables de ined in he p e ious chap e . The s a is ical cha ac e is ics o andom a iables G and Q a e de ined in Table I and a e unc ions o INTERNATIONAL JOURNAL OF MECHANICS DOI: 10.46300/9104.2020.14.14 Volume 14, 2020 ISSN: 1998-4448 112 pa ame e δ as men ioned below, see also [53]. The s a is ical cha ac e is ics o he esis ance unc ion R= y· ·b a e de ined in Table I. P ac ically Eq. (13) can be e alua ed in he o m ( ) 0<−= QDPP , (19) whe e dis ance D=R-G is conside ed as a andom a iable wi h shi ed logno mal p obabili y densi y unc ion Eq. (6). A i hme ic mean µ D, s anda d de ia ion σ D and s anda d skewness aD a e exp essed by he ollowing equa ions: GRD µµµ −= , (20) 22 GRD σσσ += , (21) 3 3 D R RD aa σ σ ⋅= , (22) The use o he shi ed logno mal p obabili y densi y unc ion o DRG is jus i ied simila ly as o R in chap e II. I can be shown ha i he shi ed logno mal p obabili y densi y unc ion Eq. (6) is sui able o R, hen i is also sui able o D, because a lowe alue o skewness (i σ D> σ R hen aD<aR) leads o a be e app oxima ion acco ding o Eq. (6). The p obabili y o ailu e Eq. (19) can be compu ed nume ically om he in eg al: ( ) ( ) ∫ ∞ ∞− Φ= yyyP QD d ϕ , (23) whe e ϕ Q(y) is Gumbel p obabili y densi y unc ion, ΦD(y) is cumula i e dis ibu ion unc ion o he shi ed logno mal dis ibu ion and y deno es a gene al poin o he obse ed o ce a iable, h ough which bo h a iables D and Q a e exp essed. In eg a ion in Eq. (23) is pe o med nume ically by Simpson’s ule using mo e han en housand in eg a ion s eps. Nume ical in eg a ion Eq. (23) allows ela i ely accu a e compu a ion o e y small alues o P in si ua ions whe e compu a ion using Mon e Ca lo nume ical simula ion me hods is ex emely nume ically demanding o impossible. The me hodology o he e i ica ion o s anda d design eliabili y is desc ibed in de ail, o example, in [53]. Mean alues µ G, µ Q and s anda d de ia ions σ G, σ Q o andom a iables G and Q a e unc ions o cha ac e is ic alues Gk and Qk acco ding o Eq. (24) and Eq. (25), whe e δ is he pa ame e o he s udy. kk k QG Q + = δ , (24) dkk RQG =⋅+⋅ 5.135.1 , (25) whe e Rd=355E6·16E-3·80E-3=454.4E3N is he design esis ance acco ding o EUROCODE 3. The alues 1.35 and 1.5 a e pa ial sa e y ac o s o pe manen and long- e m a iable load ac ion [53]. Table VII con ains he esul s o p obabilis ic analysis compu ed in ela ion o pa ame e δ , which has a s ep o 0.1. In Table VII minimum P occu s o δ =0.2 and maximum o δ =1.0. I is e iden ha he p opo ions be ween pe manen and long- e m a iable load ac ion s ongly in luence P . One way o op imize P o di e en load combina ions is he calib a ion o pa ial sa e y ac o s. Howe e , his would equi e a mo e complex s anda d assessmen , which is deba able. TABLE VII RESULTS OF FAILURE PROBABILITY ANALYSIS δ µ G σ G µ Q σ Q P 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 336.6 299.6 263.4 228.0 193.4 159.4 126.2 93.7 61.8 30.6 0.0 33.7 30.0 26.3 22.8 19.3 15.9 12.6 9.4 6.2 3.1 0.0 0.0 20.0 39.5 58.6 77.3 95.7 113.6 131.2 148.4 165.2 181.8 0.0 7.0 13.8 20.5 27.1 33.5 39.8 45.9 51.9 57.8 63.6 0.000439 0.000068 0.000022 0.000029 0.000065 0.000135 0.000248 0.000406 0.000604 0.000839 0.001105 IV. GLOBAL SENSITIVITY ANALYSIS The aim o global sensi i i y analysis is o quan i y he e ec s o inpu andom a iables on he limi s a e unc ion aking in o accoun he whole dis ibu ion o andom a iables. One app oach o es ima ing hese e ec s is h ough he applica ion o C amé - on Mises indices [44]. The adap a ion o C amé - on Mises indices o he analysis o whole dis ibu ion o andom a iables can be pe o med using Eq. (19) o Eq. (23). The es ima ion o he i s -o de indices is based on he compu a ion o p obabili y Φ DQ( τ ) and condi ional p obabili y ( ) τ DQ Φ when inpu Xi is ixed [44]. Inpu andom a iables G, Q, y, 2, b a e assumed o be s a is ically independen . ( ) ( ) ττ ≤−=Φ QDP DQ , (26) ( ) ( ) i DQ XQDP ττ ≤−=Φ , (27) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) τ ττ ττ DQ RDQDQ DQ DQ i E SΦ Φ−Φ Φ−Φ =∫d 1 2 . (28) whe e he C amé - on Mises dis ance is ( ) ( ) ( ) [ ] ( ) τττ DQ R DQ DQCVM ED ΦΦ−Φ= ∫ d 2 . (29) Es ima es o Eq. (26) and Eq. (27) a e compu ed by nume ical in eg a ion. The mean alue E[⋅] is es ima ed using INTERNATIONAL JOURNAL OF MECHANICS DOI: 10.46300/9104.2020.14.14 Volume 14, 2020 ISSN: 1998-4448 113 500 s eps o he LHS me hod. Highe -o de sensi i i y indices Sij, Sijk, e c., a e compu ed simila ly, howe e , by ixing wo, h ee, e c., inpu a iables [43]. I wo o he h ee andom a iables o R Eq. (4) a e ixed, hen he app oxima ion using he shi ed logno mal p obabili y densi y unc ion is no needed and he Gauss p obabili y densi y unc ion is used. Sensi i i y indices Si, Sij, Sijk, e c., a e based on Hoe ding decomposi ion and hus he sum o all indices mus be equal o one [43]. I can be no ed ha he nume ical complexi y o he es ima es o C amé - on Mises indices can be educed by using polynomial chaos expansions [52] as p esen ed in [63]. In he s udy p esen ed he e, i e inpu andom a iables yield i e i s -o de sensi i i y indices, and his leads o a o al o 25-1 sensi i i y indices; he e o e, 31 sensi i i y indices a e es ima ed. Compu e pa allel p ocessing can be employed o e alua e sensi i i y indices using he LHS me hod. The compu a ion o a sensi i i y index is a andom ealiza ion o his index when he LHS me hod is used. The use o new (pseudo-) andom numbe s (new andom ealiza ions o inpu andom a iables) gi es a new andom ealiza ion o he sensi i i y index. The heo e ically accu a e alue o he sensi i i y index can be ob ained ( heo e ically) ei he (i) using an in ini e numbe o s eps o he LHS me hod o (ii) by e alua ing an in ini e numbe o andom ealiza ions o he sensi i i y index (using a ini e numbe o s eps o he LHS me hod o compu e one ealiza ion o he sensi i i y index). In app oach (ii), he p ecise alue o he sensi i i y index can be ob ained as he a i hme ic mean ( heo e ical) o an in ini e numbe o andom ealiza ions o he sensi i i y index. Al hough we do no ha e he possibili y o pe o m an in ini e numbe o andom ealiza ions o sensi i i y indices, a ini e numbe can be ealized, whe e each index can be e alua ed on one CPU co e. The ad an age o app oach (ii) is ha he andom ealiza ions o sensi i i y indices can be compu ed pa allelly on he compu e . In p ac ice, he es ima e o each index is compu ed using eigh andom ealiza ions, which a e compu ed on eigh CPU co es o eigh di e en se s o (pseudo-) andom numbe s (LHS uns). The esul ing es ima e o sensi i i y indices is compu ed as he a i hme ic mean o hese eigh es ima es, esul ing in highe accu acy. The sum o all nume ically es ima ed indices is 1.000664, which is one o he indica o s o su icien accu acy o he esul s. C amé - on Mises sensi i i y indices a e compu ed o δ =0.5 and δ =0.7, see Fig. 7 and Fig. 8. The i s -o de indices accoun o mo e han 50% o he in luence on he ou pu dis ibu ion in bo h cases. The esul s show he dominan e ec o long- e m a iable load ac ion Q and yield s eng h y. The long- e m a iable load ac ion Q has a dominan e ec e en in he case o δ =0.5, Gk =Qk = 159.4 kN, see Fig. 7. The cause o he dominance is he a ia ion coe icien o Q, which has a alue o 0.35, while G has a a ia ion coe icien o 0.1. Ano he cause o he dominance o Q is Gumbel p obabili y densi y unc ion, which is used in p obabili y heo y and s a is ics o model he dis ibu ion o he maximum numbe o samples o di e en dis ibu ions [55, 56]. In he case o δ =0.7, he dominance o Q is mo e p onounced in bo h he i s -o de e ec s and in in e ac ions wi h o he a iables, see Fig. 8. The second mos in luen ial a iable is he yield s eng h, see Fig. 7 and Fig. 8 Con e sely, he specimen wid h b has p ac ically no signi ican in luence. In oducing b as a de e minis ic a iable anywhe e in he domain would ha e a much smalle e ec on limi s a e dis ibu ion han i we did he same wi h any o he inpu andom a iable. Fig. 7 C amé - on Mises indices o δ =0.5 Sobol’s decomposi ion is a p o en and widely used me hod o global sensi i i y analysis wi h many use ul applica ions, see e.g. [64-67]. The sensi i i y analysis esul s ha e been used as he basis o decision making on he modi ica ion o echnological p ocedu es o manu ac u ing beams [68]. Al hough hese a e me hods o di e en ypes, he compa ison o quan ile-o ien ed sensi i i y analysis and Sobol' sensi i i y analysis showed ha he e a e no la ge di e ences be ween he o de o impo ance o he inpu a iables, bu ce ain di e ences in he size o he indices exis [42]. I can be discussed whe he C amé - on Mises indices a e sui able so s udying s uc u al eliabili y. Es ima es o C amé - on Mises indices a e based on in eg a ion o e dΦ DQ( τ ) using Eq. (26), (27), i.e., o all τ belonging o R. As a esul o in eg a ion o e all τ , change in he mean alue o D o Q is no e lec ed in he alues o he indices. The C amé - on Mises sensi i i y analysis akes in o accoun he whole dis ibu ion o andom a iables. Thus, C amé - on Mises indices do no examine he e ec s on one P alue, bu examine he e ec s on all P alues om all τ . INTERNATIONAL JOURNAL OF MECHANICS DOI: 10.46300/9104.2020.14.14 Volume 14, 2020 ISSN: 1998-4448 114 Fig. 8 C amé - on Mises indices o δ =0.7 Con as -based sensi i i y indices [37], which examine he e ec s o inpu a iables on one a ge P and a e di ec ly ela ed o he assessed eliabili y, do no ha e his sho coming [38-39]. Con as -based sensi i i y indices belong o he ca ego y o so-called eliabili y-o ien ed sensi i i y analysis [69]. I can be no ed ha con as -based sensi i i y indices [37-39] can be w i en in a e y anspa en o m Eq. (30), bu only i he alues o P , P Xi e c. a e e y small. I can be no ed ha Eq. (30) has no in eg a ion o e τ , espec i ely, is s ill ze o. ( )( ) ( ) ( )( ) ( ) ( )( ) ,small e y o only ,1,1 1 11 0 0 2 0 0 0 XZ Z Z XZ Z i i PEco E EEE P XPEP C i i < < < < <= − = − ≈ . (30) whe e Z=D-Q, i Z<0 hen ailu e occu s, o he wise he e is no ailu e, and co is Pea son co ela ion coe icien . Nume ical expe ience wi h he LHS me hod shows ha he con e gence o C amé - on Mises indices [43] is much be e han he con e gence o p obabili y-o ien ed con as sensi i i y indices [37]. The sequence in Eq. (29) is s ongly con e gen as Mon e Ca lo uns go o in ini y and is asymp o ically Gaussian. In con as , he con e gence o Ci Eq. (30) and all o he highe o de con as -based sensi i i y indices is e y slow and equi es an ex eme numbe o sampling uns o di ec in eg a ion calcula ions. I can be no ed ha algo i hms o e ec i e es ima ions o con as -based sensi i i y indices [37] using polynomial chaos expansions [52] o neu al ne wo ks [70] would be desi able so ha mo e complex p oblems could be sol ed. V. CONCLUSION The me hodology o s ochas ic eliabili y analysis, which consis s o he p obabilis ic analysis o ailu e and global sensi i i y analysis, is desc ibed in his a icle. Global sensi i i y analysis is based on he compu a ion o he C amé - on Mises indices whils aking in o accoun he whole dis ibu ion o andom a iables. In he p esen ed case s udy, esis ance is a unc ion o he p oduc o h ee andom a iables wi h Gauss p obabili y densi y unc ions (yield s eng h, specimen hickness, specimen wid h), load ac ion is a unc ion o he sum o wo andom a iables (pe manen load ac ion, a iable ac ion). A i hme ic mean, s anda d de ia ion, and s anda d skewness o esis ance a e calcula ed analy ically using he mean alues and s anda d de ia ions o yield s eng h, specimen hickness and specimen wid h. I is shown ha he use o he shi ed logno mal p obabili y densi y unc ion is sui able o he app oxima ion o he ela i e equency o andom esis ance. This was con i med by compa ing he esul s o s a is ical and p obabilis ic analyses wi h he esul s o nume ical in eg a ion and goodness-o - i es s. Eigh de e minis ic alues o he load- ca ying capaci y a e selec ed and he p obabili y ha he andom esis ance is less han he selec ed alues is compu ed eigh imes. The p obabili y ob ained using he shi ed logno mal p obabili y densi y unc ion is in pe ec ag eemen wi h he esul o nume ical in eg a ion, which om an enginee ing poin o iew can be conside ed as a p ac ically accu a e (bu also e y nume ically demanding) solu ion. Gauss p obabili y densi y unc ion did no show such ag eemen . Goodness-o - i es s ha e shown ha he use o he shi ed logno mal p obabili y densi y unc ion o he app oxima ion o he p obabili y densi y o esis ance is be e han using he Gauss Logno mal, Gamma o Be a p obabili y densi y unc ion. Al e na i ely, he He mi e p obabili y densi y unc ion can also be conside ed, which could be explo ed in he u u e. These conclusions we e made o he case s udy p esen ed he e. Compu a ion o he p obabili y o ailu e is based on he app oxima ion o esis ance using a h ee-pa ame e logno mal dis ibu ion, which is e e ed o as he shi ed logno mal p obabili y densi y unc ion. The p esen ed solu ion is highly e ec i e o he case s udy p esen ed in his pape . In he p esen ed case s udy, he ad an ages o he p esen ed me hod a e he e y as and accu a e es ima ion o he p obabili y o ailu e. The s ochas ic model based on he shi ed logno mal p obabili y densi y unc ion and Gumbel p obabili y densi y unc ion gi es e y good accu acy o he es ima e o he p obabili y o ailu e and a he same ime has ela i ely low demands on CPU ime o he compu e . Bo h o hese p ope ies a e desi able in eliabili y-o ien ed sensi i i y INTERNATIONAL JOURNAL OF MECHANICS DOI: 10.46300/9104.2020.14.14 Volume 14, 2020 ISSN: 1998-4448 115