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 .
Acknowledgemen
This p ojec was pe o med wi h he inancial suppo o
he G an Agency SR (VEGA 1/0265/16).
Re e ences
[1] IAEA Sa e y Guide 50-SG-QA6, Re . 1, "Quali y
Assu ance in he Design o Nuclea Powe Plan s”, 1995.
[2] IAEA Sa e y S anda ds, Sa e y in Nuclea Powe Plan
Si ing, A Code o P ac ice, No. SS-50-C-S, Vienna,
No embe 1978.
[3] IAEA Sa e y Guide 50-SG-QA6, Re . 1, "Quali y
Assu ance in he Design o Nuclea Powe Plan s”, 1995.
[4] IAEA Sa e y Guide SO-SG-Sl, Re . 1, "Ea hquakes
and Associa ed Topics in Rela ion o Nuclea Powe Plan
Si ing", 1991.
[5] IAEA, Sa e y o Nuclea Powe Plan s: Design, Sa e y
S anda ds Se ies No. NS-R-1, IAEA, Vienna, 2000.
[6] IAEA Sa e y S anda ds, Ex e nal E en s Excluding
Ea hquakes in he Design o Nuclea Powe Plan s, No.
NS-G-1.5, Vienna, Feb ua y 2003.
[7] IAEA Sa e y S anda ds, Ex eme ex e nal e en s in he
design and assessmen o nuclea powe plan s, No.
IAEA-TECDOC-1341, Vienna, Ma ch 2003.
[8] IAEA Sa e y S anda ds, Me eo ological E en s in Si e
E alua ion o Nuclea Powe Plan s, No. NS-G-3.4,
Vienna, Feb ua y 2003.
[9] IAEA Sa e y S anda ds, Ad anced nuclea plan
design op ions o scope wi h ex e nal e en s, IAEA-
TECDOC-1487, Vienna, Feb ua y 2006.
[10] IAEA Sa e y S anda ds, Me eo ological and
Hyd ological Haza ds in Si e E alua ion o Nuclea
Ins alla ions, No. SSG-18, Vienna 2011
[11] NRC, RG 1.200, An app oach o de e mining he
echnical adequacy o p obabilis ic isk assessmen esul s
o isk-in o med ac i i ies, U.S. Nuclea Regula o y
Commission, Washing on, DC. 2009.
[12] NUREG-1150. Se e e Acciden Risks: An
Assessmen o Fi e US Nuclea Powe Plan s, Final
Summa y Repo , Vol.1 and 2, US NRC, 1990.
[13] NUREG/CR-4839, “Me hods o Ex e nal E en
Sc eening Quan i ica ion: Risk Me hods In eg a ion and
E alua ion P og am (RMIEP) Me hods De elopmen ,”
Repo , Sandia Na ional Labo a o ies and U.S. Nuclea
Regula o y Commission, 1992.
[14] UJD SR, The s ess es s o Nuclea Powe Plan s
Slo akia, B a isla a, Sep embe 2011.
[15] KRÁLIK, J. Sa e y and Reliabili y o Nuclea Powe
Buildings in Slo akia. Ea hquake-Impac -Explosion.
Monog aph. Edi ion STU B a isla a, 305 p. 2009.
[16] SALAJKA, V. P. HRADIL, J. KALA, Assess o he
Nuclea Powe Plan S uc u es Residual Li e and
Ea hquake Resis ance, In P oc. The Second In e na ional
Con e ence on Enginee ing and Technology Inno a ion
(ICETI 2012), Kaohsiung, Taiwan, No embe 02-06,
2012, pp.
[17] ANTUCHEVICIENE,J. Z. KALA, M. MARZOUK
and E. R. VAIDOGAS, Decision Making Me hods and
Applica ions in Ci il Enginee ing, Hindawi Publishing
Co po a ion, Ma hema ical P oblems in Enginee ing,
Volume 2015, ID 160569, 3 pages, h p://dx.doi.o g/
10.1155/2015/160569
[18] ČAJKA, R. and M. KREJSA, Measu ed Da a
P ocessing in Ci il S uc u e Using he DOP oC Me hod.
Ad anced Ma e ials Resea ch. Zu ich, Swi ze land:
T ans Tech Publica ions, 2014. Vol. 859, pp. 114-121 (8 p),
ISSN 1662-8985, DOI:10.4028/www.scien i ic.ne / AMR.
859.114.