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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
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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.
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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
)
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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
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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
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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.