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FULLY STOCHASTIC NONLINEAR ANALYSIS OF SLENDER
REINFORCED CONCRETE COLUMN
Filip ŠMEJKAL1, Radomí PUKL1, Jan ČERVENKA1
1 Če enka Consul ing s. .o., Na Hřebenkách 55, P ague, Czech epublic
[email p o ec ed], [email p o ec ed], [email p o ec ed]
DOI: 10.35181/ ces-2020-0007
Abs ac . The ollowing a icle p esen s nonlinea
esis ance assessmen o a slende ein o ced conc e e
column using comme cial FEM so wa e ATENA.
Fu he mo e, h ee di e en app oaches a e used o
de e mine design alue o esis ance. Fi s ly, he mos
commonly used me hod o pa ial sa e y ac o s is
desc ibed. Secondly, me hod ECoV (es ima e o coe icien
o a ia ion) is p esen ed, which is one o he possible
op ions used in ib Model Code o asses design esis ance
o s uc u es using nonlinea analysis. E en ually, a ully
p obabilis ic analysis is pe o med using comme cial
so wa e package SARA.
Keywo ds
Non-linea analysis, sa e y o ma s, eliabili y.
1. In oduc ion
Non-linea s uc u al analysis plays an impo an ole
while assessing eliabili y o exis ing, o e en newly
designed conc e e (conc e e-s eel) s uc u es. Al hough he
simpli ied linea elas ic analysis is a powe ul ool and
allows us ela i ely quickly design dimensions o a desi ed
s uc u e, i we wan o ge a deepe insigh in o he eal
s uc u al beha iou unde ce ain load condi ions, he
linea elas ic analysis quickly becomes insu icien
because i doesn’ accoun o geome ical (equilib ium on
de o med s uc u e) and mos impo an ly ma e ial
(conc e e c acking and c ushing, ein o cemen yielding,
e c.) nonlinea i ies. I also wo h men ioning ha elas ic
dis ibu ion o in e nal o ces in s a ically inde e mina e
s uc u es is usually close o eali y only unde e y low
load le els, which is in con a y wi h design p ocedu e
desc ibed e.g. in Eu ocode 2 (EC2) [1], whe e he elas ic
dis ibu ion o in e nal o ces is used o design he c oss-
sec ional dimensions (and ein o cemen ) unde an
assump ion o plas ic ma e ial beha iou . Ne e heless,
al hough his design p ocedu e doesn’ p ope ly e lec eal
beha iou o he s uc u e, i has been p o ed by many
yea s o expe ience ha i p o ides conse a i e designs.
On he o he hand, i one wan s o simula e he eal
beha iou o gi en ein o ced conc e e s uc u e, i is
necessa y o accoun o nonlinea i ies. One o he possible
solu ions is o use comme cial FEM so wa e ATENA [2],
which can model disc e e ein o cemen and uses ac u e-
plas ic ma e ial model o simula e he eal conc e e
beha iou . P inciples o he model can be cha ac e ized by
ollowing ea u es:
• Smea ed c acks wi hin ini e elemen s;
• C ack band con ol o s ain localiza ion
including e ec s o c ack o ien a ion and
elemen shape unc ion;
• Fixed c ack model;
• Two ac u e ailu e modes a e conside ed on he
c ack ace, Mode I due o no mal s ess ac ion
(exponen ial law o c ack opening con olled by
ac u e) and Mode II due o shea s ess (shea
de o ma ion con olled by a shea ac o and
shea s eng h depending on agg ega e
in e lock);
• Reduc ion o comp essi e s eng h due o
damage by c acks;
• Mené ey-Willam [3] plas ici y o mula ion o
conc e e in comp ession wi h non-associa ed
low ule.
Mo e de ailed in o ma ion abou he ma e ial model can be
ound in [4].
The pu pose o his pape is o compa e h ee di e en
me hods o global design esis ance e alua ion, me hod o
pa ial sa e y ac o s, me hod ECoV (es ima e o
coe icien o a ia ion) and ull p obabilis ic analysis.
Resul s will be p esen ed on he ela i ely simple
ein o ced conc e e s uc u e – slende column.
2. Global Sa e y Fo ma
In s anda d design p ocedu e, he ollowing condi ion has
o be ul illed
dd
F
R<, (1)
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whe e d
F
s ands o design ac ion and d
R s ands o
design esis ance. Mos commonly used design me hod
desc ibed e.g. in Eu ocode 2 is based on ul illing
condi ion (1) in all he c i ical c oss-sec ions o he gi en
s uc u e. d
F
is han design in e nal o ce (no mal o ce,
bending momen , e c.) calcula ed by means o linea
elas ic analysis on he gi en s uc u e loaded by p esc ibed
load combina ions ma gined by pa ial sa e y ac o s (e.g.
G
γ
o pe manen load, Q
γ
o imposed load,
P
γ
o
p es essing load, e c.) and d
R is esis ance o he selec ed
c oss-sec ion o he s uc u e o he ac ing in e nal o ce,
which is usually de e mined unde plas ic ma e ial
condi ions using ma e ial sa e y ac o s
M
γ
. In his
p ocedu e we assume ailu e p obabili ies o sepa a e
ma e ials in he c i ical c oss-sec ions, bu he ailu e
p obabili y o he en i e s uc u e emains unknown. On
he o he hand, in non-linea analysis, d
Ris design global
esis ance, i.e. se o o ces ep esen ing an imposed load
(load combina ion) which lead o ailu e o some pa o
he s uc u e. Unlike in sec ional design, he global
esis ance accoun s o in e ac ion o he whole s uc u e
and does no only asses speci ic c oss-sec ions. I can be
exp essed as [5]
m
d
R
R
R
γ
=, (2)
whe e m
R is he mean esis ance and
R
γ
is he global
sa e y ac o , which includes all unce ain ies and unde an
assump ion o log no mal dis ibu ion used in Eu ocode 2
i can be exp essed by means o coe icien o a ia ion o
esis ance
R
V as
()
exp
R
RR
V
γαβ
=, (3)
whe e
R
α
( 0.8
R
α
= o 0.001 p obabili y o ailu e) is
he sensi i i y ac o o esis ance and
β
( 3.8
β
= o a
e e ence pe iod o 50 yea s) is he eliabili y index. I is
impo an o no e ha coe icien o a ia ion
R
V includes
unce ain ies o a ious o igins and can be exp essed as [6]
222
R
GmRd
VVVV=++
, (4)
whe e G
V, m
V, and
R
d
Va e coe icien s o a ia ion o
associa ed andom a iables o accoun o geome y,
ma e ial and nume ical model unce ain ies.
The e a e di e en app oaches o assessmen o design
s uc u e eliabili y which di e in he le el o
app oxima ion. Th ee o hem a e b ie ly desc ibed below.
2.1. Full P obabilis ic Analysis
The ull p obabilis ic analysis is he mos a ional way o
assessing he s uc u al eliabili y. I s p incipal lies in
unning he non-linea simula ions on many samples o he
in es iga ed s uc u e, while chosen pa ame e s o he
nume ical model (ma e ial p ope ies, dimensions,
bounda y condi ions, e c.) a e sys ema ically a ied wi hin
he samples acco ding o ce ain p obabili y dis ibu ion
unc ions (PDF). The e a e di e en algo i hms o
gene a ing he samples, which a e desc ibed in de ail e.g.
in [7] and won’ be u he discussed in his pape . The
andomiza ion o chosen p obabilis ic quan i ies in
nume ical model can be ca ied ou in wo di e en ways:
(a) andom a iables, whe e he quan i y emains
cons an wi hin he sample (s uc u e), bu di e s be ween
samples and (b) andom ields, whe e he quan i y a ies
andomly in space and o cou se be ween he samples. I is
impo an o no e ha ce ain pa ame e s a e ac ually
co ela ed (e.g. in case o conc e e, he highe he
comp essi e s eng h, he highe he ensile s eng h) and
he e o e his co ela ion should be aken in o accoun o
p ope ly e lec he eali y. A e unning he nume ical
simula ions we end up wi h an a ay o esis ance alues
which can be i by chosen PDF o esis ance (e.g. in EC2
i is log no mal PDF). The only emaining s ep is o choose
he p obabili y o ailu e we wan o ob ain he esis ance
alue o . In ou case, we a e in e es ed in design alue o
esis ance, which usually co esponds o he 0.001
p obabili y o ailu e (excluding he unce ain y o ac ion
o ce). Al hough his me hod is he mos a ional and
obus way o de e mining he s uc u al design esis ance,
i is compu a ionally e y demanding (la ge numbe o
simula ions has o be pe o med). Fu he mo e, i one
wan s o ob ain esis ance alues o e y small (o e y
la ge) ailu e p obabili ies (which is he case o design
esis ance), he esul s s ongly depend on choice o PDF,
because he e is usually a lack o samples on he edges o
PDF and he esis ance alue is he e o e ex apola ed.
2.2. ECoV Me hod – Es ima e o Coe icien
o Va ia ion
ECoV is a simpli ied p obabilis ic me hod p oposed in [9].
I is based on he idea o de e mining he coe icien o
a ia ion om wo samples only and assumes logno mal
dis ibu ion o esis ance. The i s sample is calcula ed
using he mean ma e ial pa ame e s (which supposedly
co esponds o he median) and he second one uses he
cha ac e is ic pa ame e s ( ha a e supposed o yield 5%
quan ile). A e unning he non-linea analysis on bo h
samples, coe icien o a ia ion o s uc u al esis ance
can be calcula ed as
1ln
1.65
m
R
k
R
VR

=

, (5)
whe e k
R is cha ac e is ic esis ance.
The esul ing design esis ance is hen ob ained using
o mulas (2) and (3). This me hod is gene al and much less
compu a ionally demanding han ully p obabilis ic
app oach. Howe e , i is a guable i i p ope ly e lec s all
ypes o ailu e. In [6] au ho s p opose simila me hod
which equi es mo e samples han ECoV, bu is s ill much
less compu a ionally demanding han ully p obabilis ic
app oach.
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2.3. Pa ial Sa e y Fac o s
The me hod o pa ial sa e y ac o s desc ibed in mos o
he design codes can be also applied on global analysis o
ob ain design esis ance o he s uc u e. The design alues
o ma e ial pa ame e s d
a e calcula ed as /
dkM
γ
=,
whe e k
is cha ac e is ic alue and
M
γ
is ma e ial sa e y
ac o , which can be calcula ed as [6]
()
exp 1.64
M
RR m
VV
γαβ
=−
(6)
o ein o cing s eel and as
()
1.15 exp 1.64
MRRm
VV
γαβ
=−
(7)
o conc e e. An addi ional ac o 1.15 in (7) has been
in oduced o accoun o he lowe conc e e s eng h in
eal s uc u es han in ca e ully cu ed expe imen al
cylinde s.
Speci ic alues o coe icien s o a ia ion which lead
o well-known alues 1.5
C
γ
= o conc e e and 1.15
S
γ
=
a e using (4) in (6) and (7) a e shown in Tab. 1.
Tab. 1: S a is ical ep esen a ion which leads o he pa ial sa e y
ac o s in Eu ocode 2.
Type o
unce ain y Rein o cing s eel Conc e e
R
d
V (model) 2.5% 5%
G
V (geome y) 5% 5%
m
V (ma e ial) 4% 15%
Conside ing ha design alues calcula ed by means o
pa ial sa e y ac o s ep esen ex emely low alues o
ma e ial p ope ies, his me hod could lead o dis o ed
ailu e modes (meaning ha ailu e mode o he s uc u e
can be un ealis ic due o he ex emely low alues o
ma e ial p ope ies which a e unlikely o occu ). On he
o he hand, yea s o expe ience p o ed, ha his me hod
gi es mos ly conse a i e and he e o e sa e esul s.
3. Case S udy
The h ee abo e men ioned me hods o assessing he
global s uc u al eliabili y will be p esen ed on he case
s udy o axially loaded ein o ced conc e e slende
column. The loading scheme and basic dimensions o he
column a e shown in Fig. 1. The de ailed d awing o
ein o cemen can be u he seen in Fig. 2.
3.1. Nume ical Model
3D nume ical model o p esen ed case s udy wi h disc e e
ein o cemen was c ea ed using comme cial FEM
so wa e ATENA and i s de ails a e desc ibed in he
ollowing subchap e s.
1) Ma e ials
To in es iga e he in luence o conc e e s eng h, he
s uc u al beha io is simula ed wi h wo conside ably
di e en classes o conc e e, C50/60 and C16/20. The
ein o cemen s eel is B500B. The ollowing ma e ial
p ope ies a e used (Tab. 2, Tab. 3 and Tab. 4).
Tab. 2: Ma e ial p ope ies o conc e e class C50/60 om EC2.
C50/60 Mean Cha ac e is ic Design
Ec [MPa] 37277 35650 32102
ν
0.2 0.2
0.2
c [MPa] -58 -50 -35.24
[MPa] 4.1 2.9 2.04
G [N/m] 102 72.5 51.1
ε
cp -0.000897 -0.00111
-0.00156
Tab. 3: Ma e ial p ope ies o conc e e class C16/20 om EC2.
C16/20 Mean Cha ac e is ic Design
Ec [MPa] 28608 25331 22803
ν
0.2 0.2
0.2
c [MPa] -24 -16 11.27
[MPa] 1.9 1.3 0.916
G [N/m] 47.5 32.5 22.9
ε
cp -0.00104 -0.00132
-0.001507
E
c is modulus o elas ici y,
ν
is Poisson’s a io, c is
comp essi e s eng h, is ensile s eng h, G is ac u e
ene gy and
ε
cp is plas ic s ain a comp essi e s eng h
le el. I is impo an o no e, ha he mean modulus o
elas ici y Ecm can be acco ding o EC2 calcula ed as
()
0.3
22000 /10
cm cm
E =, bu EC2 doesn’ say any hing
abou he cha ac e is ic alue o E modulus. Ac ually, i
p oposes ha he mean alue should be always used.
Howe e , in case o he analysis ha can s ongly depend
on he alue o E modulus (e.g. because o buckling
phenomena), i is necessa y o use Ek and Ed di e en om
Em. The simples in e p e a ion o EC2 is o apply analogy
o he o mula used o de e mine he mean alue o E and
calcula e cha ac e is ic alue o modulus o elas ici y as
()
0.3
22000 /10
ck ck
E = and design alue as
()
0.3
22000 /10
cd cd
E =, which is done in Tab. 2 and Tab.
3. Fu he mo e, o compa e esul s ob ained om ull
p obabilis ic analysis and om ECoV wi h me hod o
pa ial sa e y ac o s, we only need o accoun o ma e ial
unce ain y in ma e ial sa e y ac o s. Using o mulas (4),
(6), and subs i u ing 0
G
V= and 0
Rd
V= gi es us
()
()
exp 1.64
exp 0.8 3.8 0.04 1.64 0.04 1.058
SRRm
VV
γαβ
=−=
=××−×=
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o s eel ein o cemen sa e y ac o and using (7)
analogically gi es us
()
()
1.15 exp 1.64
exp 0.8 3.8 0.15 1.64 0.15 1.419
CRRm
VV
γαβ
=× − =
=××−×=
o conc e e sa e y ac o .
Tab. 4: Ma e ial p ope ies o ein o cemen s eel B500B om EC2.
B500B Mean Cha ac e is ic Design
Es [MPa] 200000 200000 200000
y [MPa] 550 500 472.8
ε
lim 0.05 0.05
0.05
whe e y is he yield s eng h and
ε
lim is he limi s ain.
2) Loading
ATENA uses inc emen al me hod o applying loads by
means o so called in e als. Loading o ou speci ic model
is di ided in o wo in e als:
• In e al 1: deadload (impo an due o he
ho izon al o ien a ion o column, conc e e
densi y 2300 kg/m3); 1 s ep.
• In e al 2: axial displacemen load; 0.1 mm s eps
un il ailu e.
3) Mesh sensi i i y analysis
To ensu e ha su icien ly accu a e esul s a e ob ained
om he analysis, wo di e en mesh g ids we e used.
Bo h use linea solid hexahed al elemen s. The coa se
mesh uses ini e elemen s wi h dimensions 50×50×23 mm
(x, y, z) and he ine mesh uses ini e elemen s wi h
dimensions 30×50×12.5 mm. The mesh g id wi h
coo dina e sys em o ien a ion can be seen in Fig. 3. Mesh
sensi i i y analysis was pe o med only o conc e e class
C50/60 and he esul ing load-displacemen (L-D)
diag ams a e shown in Fig. 4. I is clea ha o his case
s udy coa se mesh can be used because he esul s di e
only sligh ly and he compu a ional ime dec eases
conside ably.
Fig. 1: Schema ic d awing o slende column.
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Fig. 2: Rein o cemen o slende column.
Fig. 3: Fini e elemen mesh g id (coa se).
Fig. 4: Compa ison o L-D diag ams o coa se and ine mesh.
3.2. Full P obabilis ic Analysis
Full p obabilis ic analysis o he case s udy is pe o med
using comme cial so wa e SARA [7, 8], which is used o
andomize ma e ial pa ame e s and u he coope a es wi h
ATENA, whe e he simula ions a e un. The andomized
ma e ial pa ame e s a e modulus o elas ici y o conc e e
Ec, conc e e ensile s eng h , conc e e comp essi e
s eng h c, conc e e ac u e ene gy G and conc e e plas ic
s ain when he comp essi e s eng h is eached,
ε
cp. All o
he andomized pa ame e s a e assumed o ha e no mal
dis ibu ion wi h gi en mean alues and s anda d
de ia ions calcula ed such ha 5% quan ile o he esul ing
p obabili y dis ibu ion unc ion ep esen s he
cha ac e is ic alue (Tab. 2 and Tab. 3). This app oach is
easonable o comp essi e and ensile s eng h o
conc e e and ques ionable o es o he pa ame e s, bu
di e en app oach would equi e mo e expe imen al da a.
Fu he mo e, some o he pa ame e s a e co ela ed in
eali y, which is aken in o accoun by se ing up he
s a is ical co ela ion ma ix (Tab. 5).
Tab. 5: S a is ical co ela ion ma ix (symme ical).
Ec
c G
ε
cp
Ec 1 0 -0.7 0 0
1 -0.5 0.8 0
c 1 0 -1
G 1 0
ε
cp
1
Co ela ion measu es he deg ee o s a is ical
associa ion be ween wo a iables. In ou case co ela ion
measu es he deg ee o linea i y o he ela ionship. The

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highe he alue o s a is ical co ela ion is, he mo e
co ela ed he quan i ies a e. Nega i e co ela ion means
ha he highe one quan i y is, he lowe is he second one
(and ice e sa). The simula ion is pe o med on 60
samples (see Fig. 5, analogical o all andomized
pa ame e s o C50/60 and C16/20).
Fig. 5: No mal dis ibu ion o comp essi e conc e e s eng h o 60
samples, C50/60 [10].
A e unning ATENA non-linea analysis o all he
gene a ed samples ( o bo h conc e e classes) we ge
ollowing L-D diag ams (Fig. 6 and Fig. 7) and eliabili y
his og ams (Fig. 8 and Fig. 9). Finally, we can ob ain he
global esis ance alue o selec ed p obabili y o ailu e,
which in ou case co esponds o 0.001 (design esis ance)
and is equal o ,,500.973 MN
xp obc
R= o conc e e class
C50/60 and ,,160.347 MN
xp obc
R= o conc e e class
C16/20.
Fig. 6: Se o L-D diag ams o conc e e class C50/60.
Fig. 7: Se o L-D diag ams o conc e e class C16/20.
Fig. 8: Reliabili y his og am, logno mal PDF, C50/60 [10].
Fig. 9: Reliabili y his og am, logno mal PDF, C16/20 [10].
3.3. ECoV Me hod
To de e mine design esis ance using ECOV me hod, only
wo samples a e equi ed o each conc e e class. Using he
mean and cha ac e is ic ma e ial p ope ies (Tab. 2 and
Tab. 3) in ou nume ical model lead o he ollowing
esis ance alues. Fo conc e e C50/60
,,50 1.099 MN
xmc
R= and ,,50 0.987 MN
xkc
R=, o
conc e e C16/20 ,,16 0.648 MN
xmc
R= and
,,16 0.466 MN
xkc
R=. To de e mine design sa e y ac o ,
one has o apply o mula (2) o de e mine coe icien o
a ia ion:
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,50
1 1.099
ln 0.0651
1.65 0.987
Rc
V
==

 ,
,16
1 0.648
ln 0.199
1.65 0.466
Rc
V
==

 ,
hen (3) o de e mine design sa e y ac o :
()
,50 exp 0.8 3.8 0.0651 1.219
Rc
γ
=××=
,
()
,16 exp 0.8 3.8 0.199 1.831
Rc
γ
=××=
,
and inally (5) o e alua e he esul ing design esis ance:
,,50
1.099 0.901MN
1.219
dECOVc
R== ,
,,16
0.648 0.354 MN
1.831
dECOVc
R== .
3.4. Pa ial Sa e y Fac o s Me hod
This me hod applies sa e y ac o s on ma e ial pa ame e s
be o e he non-linea analysis (Tab. 2, Tab. 3 and Tab. 4)
and conside s he esul ing esis ance alue as design one.
The ollowing esul s we e ob ained o each conc e e
class, ,,500.758 MN
d pa c
R= o C50/60 and
,,160.349 MN
d pa c
R= o C16/20.
3.5. Resul s and Discussion
The inal summa y o esul s ob ained by applying abo e
desc ibed me hods is shown in Fig. 10 o C50/60 and Fig.
11 o C16/20.
Fig. 10: Compa ison o h ee design esis ance assessmen me hods,
C50/60.
Fig. 11: Compa ison o h ee design esis ance assessmen me hods,
C16/20.
As is clea om he diag ams, in case o highe s eng h
conc e e (C50/60), he esul ing alues o design esis ance
ob ained by a ious me hods di e conside ably, while o
low s eng h conc e e, he esul ing design esis ances
ma ch pe ec ly. This esul could lead o conclusion, ha
in case o highe s eng h conc e e, he pa ial sa e y
ac o s me hod is unnecessa ily conse a i e and use o
ECoV, o possibly ull p obabilis ic analysis would esul
in mo e economical designs. On he o he hand, i s ill
emains unclea why in case o highe s eng h conc e e
he design esis ance ob ained by me hod o pa ial sa e y
ac o s di e s ha much om he alues ob ained by o he
wo me hods. One o he possible explana ions can be ha
EC2 de ines mean comp essi e s eng h alue as
8MPa
cm ck
=+ , which o low s eng h conc e e
classes leads o conside able highe ela i e di e ence
be ween he mean and cha ac e is ic alues and he e o e
he ull p obabilis ic analysis and me hod ECoV gi e
smalle alues o design esis ance i.e. close o he pa ial
sa e y ac o s me hod.
4. Conclusions
Th ee me hods o global design esis ance assessmen o
ein o ced conc e e s uc u es we e p esen ed whe eas
each one o hese me hods uses di e en le el o
app oxima ion. Non-linea s uc u al analysis we e
pe o med in comme cial FEM so wa e ATENA using
disc e e ein o cemen and 3D ac u e-plas ic ma e ial o
conc e e. Fi s ly, he mos obus , bu compu a ionally
demanding ull p obabilis ic analysis was pe o med on 60
samples o wo conc e e classes by means o comme cial
ool o s uc u al eliabili y assessmen , SARA, and he
design esis ance was de e mined o 0.001 p obabili y o
ailu e. Secondly, simpli ied p obabilis ic me hod ECoV,
which equi es only wo samples (mean and
cha ac e is ic), was applied o calcula e he design
esis ance. A las , mos commonly used me hod o pa ial
sa e y ac o s was applied o educe ma e ial pa ame e s o
design alues, which inpu he non-linea analysis.
E en ually, all he esul s we e compa ed in diag ams and
he di e ences we e discussed.
SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 20 | NUMBER: 1 | 2020 | JUNE
© 2020 TRANSACTIONS OF VSB - TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 45
Acknowledgemen s
The p esen ed esea ch was pa ially suppo ed by TAČR
wi hin Del a p og am, p ojec No. TF05000040 "CeSTaR-
Compu e simula ion and expe imen al alida ion-
complex se ice o lexible and e icien design o p e-
cas conc e e columns wi h inno a i e mul i-spi al
ein o cemen ".
Re e ences
[1] BS EN 1992, Eu ocode 2: Design o conc e e
s uc u es.
[2] ČERVENKA, J., L. JENDELE and V. ČERVENKA.
ATENA P og am documen a ion, Theo y. Če enka
Consul ing, www.ce enka.cz, 2019.
[3] MENETREY, P. and K.J. WILLAM. T iaxial Failu e
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[4] ČERVENKA, J. and V.K. PAPANIKOLAOU. Th ee
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Abou Au ho s
Filip ŠMEJKAL was bo n in P ague, Czech Republic. He
ecei ed his M.Sc. om Facul y o Ci il Enginee ing a
Czech Technical Uni e si y in P ague in P ague in 2017.
His esea ch in e es s include nume ical modeling o
ein o ced conc e e s uc u es.
Radomí PUKL was bo n in P ague, Czech Republic. He
ecei ed his Ph.D. deg ee om he Klokne ins i u e a
Czech Technical Uni e si y in P ague. His esea ch
in e es s include compu e simula ions o conc e e
s uc u es.
Jan ČERVENKA was bo n in P ague, Czech Republic.
He ecei ed his Ph.D. deg ee om he Uni e si y o
Colo ado in 1994. His esea ch in e es s include
de eloping nonlinea ma e ial models o conc e e.