On Ma hema ical Models o Deg ada ion P ocesses
Acco ding o ISO 16204 and ib Model Code
Ma ina Šomodíko á1 | Jiří Doležel1 | Da id Lehký1
1 In oduc ion
In he ib Model Code 2010 [1], se ice li e is de ined as
“ he pe iod in which he equi ed pe o mance o a s uc-
u e o s uc u al elemen is achie ed, when i is used o
i s in ended pu pose and unde he expec ed condi ions o
use”. I is a empo al (quan i a i e) alue ha is no a
ma e ial p ope y bu is ela ed o he abili y o ma e ials,
componen s and sys ems o main ain speci ic u ili y and
o he p ope ies a he equi ed le el unde no mal
main enance, o e a ce ain ime pe iod and unde gi en
ope a ing and en i onmen al condi ions.
The me hods o de e mining he esidual se ice li e o
new and exis ing s uc u es may be as ollows:
Based on knowledge o he se ice li e o a simila
s uc u e (wi h simila cha ac e is ics) loca ed in sim-
ila condi ions;
On he basis o accele a ed es s;
Using ma hema ical models;
Using s ochas ic me hod – a me hod using a eliabili y
model o a me hod using a combina ion o s a is ical
and de e minis ic models may be used.
In he case o he de e mina ion o se ice li e based on
he knowledge o he se ice li e o a simila s uc u e, i
is a he an es ima e due o he la ge a iabili y, especially
in he p ope ies o he ma e ials used and he in luence
o he a ious en i onmen s. Fo he de e mina ion based
on accele a ed es s, i is necessa y ha he deg ada ion
mechanisms o he ma e ial in accele a ed condi ions a e
he same as in he eal en i onmen . E en i his condi ion
is me , he lack o da a on he deg ada ion a e unde no -
mal condi ions, based on long- e m moni o ing o he con-
di ion o s uc u es and long- e m es ing, is a majo p ob-
lem in he de e mina ion o se ice li e. The e o e,
ma hema ical modelling and s ochas ic me hods, which
a e he ocus o his pape , appea o be app op ia e me h-
ods o he se ice li e assessmen . Rela ed o his issue
he u iliza ion o a s anda dized me hodology implemen ed
in he indus y wo ldwide plays an impo an ole.
A he u n o he millennium, e o s we e ini ia ed o
de elop a pla o m o du abili y design o conc e e
s uc u es ha con ained he same elemen s and
philosophy as ha o mode n s uc u al design [2]. In
ollowing ew yea s he ib Bulle in No. 34 [3] was
endo sed. The close coope a ion be ween he ib and ISO
commi ees was es ablished and based on he p inciples o
ISO 2394 [4] he ib Model Code 2010 [1] and he ISO
16204 [5] documen s we e inalized. These wo
documen s a e oday close o being iden ical when
assessing he se ice li e and du abili y o con e e
s uc u es.
2 Ma hema ical modelling o deg ada ion p o-
cesses
In mos cases, he se ice li e o a s uc u e, S, ela ed o
ORIGINAL ARTICLE
Abs ac
The main ac o s a ec ing he se ice li e o conc e e s uc u es a e he p esence o
chlo ide ions and ca bon dioxide, al e na ing os ac ion and mechanical s esses
on he s uc u e. In p ac ice, hese e ec s can be aken in o accoun by using ma h-
ema ical modelling o hese phenomena. In his con ibu ion, he au ho s ocus on
he p ocesses o chlo ide ions and ca bon dioxide (ca bona ion p ocess) di usion
h ough conc e e and he associa ed subsequen co osion o ein o cemen . Ma h-
ema ical models ecommended by he ib Model Code and ISO 16204 a e desc ibed
and he e ec o simul aneous en i onmen al and mechanical load on he se ice
li e o he s uc u e is discussed. The models a e used o s udy he esidual se ice
li e o a simple ein o ced conc e e ame b idge based on he pe cen age loss o
ein o cemen a ea analysis o e ime.
Keywo ds
ib Model Code, ISO 16204, Conc e e Ca bona ion, Chlo ide Ions Ing ess, Co osion
o Rein o cemen
Co espondence
Ing. Ma ina Šomodíko á, Ph.D.
B no Uni e si y o Technology
Facul y o Ci il Enginee ing,
Ins i u e o S uc u al Mechanics
Ve eří 331/95
60200 B no
Email: somodiko a.[email p o ec ed].cz
1 B no Uni e si y o Technology,
Facul y o Ci il Enginee ing, B no,
Czech Republic
P oceedings
in ci il enginee ing
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dis ibu ion in any medium, p o ided he o iginal wo k is p ope ly ci ed, he use is non-comme cial and no modi ica ions o adap a ions a e
made.
h ps://doi.o g/10.1002/cepa.2135 wileyonlinelib a y.com/jou nal/cepa
ce/pape s 6 (2023), No. 5
© 2023 E ns & Sohn GmbH.
1221
he du abili y limi s a e, is de ined as he ime ha elapses
om he ime he s uc u e is pu in o se ice o he ime
when he ein o cemen co osion p ocess is ini ia ed, i.e.,
S = i. In he case o pene a ion o chlo ide ions and ai
CO2 h ough he conc e e, he co osion p ocess is ini ia ed
by di usion o chlo ide ions in he conc e e o he dep h a
which he s eel ein o cemen is loca ed, o by ca bona-
ion, which lowe s he pH o he conc e e a i s con ac
wi h he s eel ein o cemen . Taking a less conse a i e
iew, he se ice li e can be de e mined as he sum o wo
ime pe iods, i.e., he ini ia ion phase, i, and he p opa-
ga ion phase, p, as S = i + p. Since he co osion o he
ein o cemen educes i s e ec i e a ea, he sa e y c i e-
ion can be de ined, o example, on he basis o eaching
a limi alue o pe cen age loss o a ea o he ein o ce-
men ba s. No e he e ha while he a ea o he ein o ce-
men dec eases, he co osion p oduc s o med ha e a
wo old o six old inc ease in olume, which leads o an
inc ease in ensile s ess in he su ounding conc e e, and
hus o he o ma ion o longi udinal con inuous c acks and
consequen spalling o he conc e e co e .
2.1 Co osion o ein o cemen
The cou se o ein o cemen co osion, which akes place
du ing he p opaga ion s age o he assumed se ice li e
o he s uc u e, i.e., a e he ini ia ion ime o ein o ce-
men depassi a ion is eached, can be modelled acco ding
o [6]. Fo he case o uni o m co osion, he ollowing
equa ions a e used o es ima e he ein o cemen diame-
e d [mm] o e ime:
𝑑(𝑡)={ 𝑑i
𝑑i−0.0116𝑖co 𝑅co (𝑡−𝑡i)
0 (1a)
o he p esc ibed condi ions o :
{
𝑡≤𝑡i
𝑡i<𝑡≤𝑡i+𝑑i
0.0116𝑖co 𝑅co
𝑡>𝑡i+𝑑i
0.0116𝑖co 𝑅co
(1b)
whe e di [mm] is he ini ial diame e o he ein o cemen ,
ico [μA/cm2] akes he co osion cu en densi y (co o-
sion a e) in o accoun and Rco [-] e lec s he ype o
co osion (uni o m o pi ing). Simila ly, he pi ing dep h
p [mm] can be de e mined o e ime o pi ing co osion
as [7]:
𝑝(𝑡)={ 0 o 𝑡≤𝑡i
0.0116𝑖co 𝑅co (𝑡−𝑡i) o 𝑡>𝑡i (2)
Assuming a hemisphe ical shape o he pi he esidual
(ne ) c oss-sec ional a ea o he co oded ba (A [mm2],
g ey a ea in Fig. 1 le ) a ime > i can be calcula ed as
[8]:
𝐴 (𝑡)=
{
π𝑑i2
4−𝐴1−𝐴2 o 𝑝(𝑡)≤√2
2𝑑i
𝐴1−𝐴2 o √2
2𝑑i<𝑝(𝑡)≤𝑑i
0 o 𝑝(𝑡)>𝑑i
(3a)
wi h
𝐴1=12[𝜃1(𝑑i
2)2−𝑎p|𝑑i
2−𝑝(𝑡)2
𝑑i|]
𝐴2=12[𝜃2𝑝(𝑡)2−𝑎p𝑝(𝑡)2
𝑑i]
𝑎p=2𝑝(𝑡)√1−(𝑝(𝑡)
𝑑i)2
𝜃1=2a csin(𝑎p
𝑑i)
𝜃2=2a csin(𝑎p
2𝑝(𝑡))
(3b)
The c oss-sec ional a ea can be calcula ed wi hou signi i-
can loss o accu acy using a simple equa ion ha as-
sumes a ci cula pi shape and a ein o cemen a ea cal-
cula ed based on he o e lap o wo ci cles o adius di (see
Fig. 1 igh ), as in [9]:
𝐴 (𝑡)={π𝑑i2
4−𝐴p o 𝑝(𝑡)≤𝑑i
0p o 𝑝(𝑡)>𝑑i (4a)
wi h
𝐴p=𝑑i2
4(𝛿−sin 𝛿)
𝛿=2a ccos(1−𝑝(𝑡)
𝑑i) (4b)
Figu e 1 A ea o ein o cing ba wi h pi ing co osion – hemisphe ical
pi (le ), ci cula pi ( igh )
2.2 Conc e e ca bona ion and chlo ide ions in-
g ess
As al eady men ioned abo e, he depassi a ion o ein-
o cemen can be caused mainly by ca bona ion p ocess
and/o by he chlo ide ions ing ess, whe e he ini ia ion
ime is de e mined on he basis o he assump ion ha he
ca bona ion on eaches he dep h a which he s eel
membe s a e loca ed o he chlo ide ion concen a ion a
he dep h o he s eel ein o cemen eaches a c i ical
alue. The ma hema ical models o ca bona ion and chlo-
ide pene a ion a e based on Fick's second law o di u-
sion:
𝜕𝐶
𝜕𝑡=𝐷𝜕2𝐶
𝜕𝑥2 (5)
whe e C is he concen a ion o he pe mean , x is he dis-
ance om he su ace, is he di usion ime and D is
gene ally he di usion coe icien , which depends on he
p ope ies o he conc e e and he en i onmen . The solu-
ion o his di e en ial equa ion is ound by he C ank p o-
cedu e using he Gaussian e o unc ion “e (.)”.
Acco ding o ISO 16204 [5], he ca bona ion dep h xc
[mm] a ime [yea s] can be calcula ed as:
𝑥c(𝑡)=𝑊𝑘√𝑡 (6)
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whe e he pa ame e W [-] akes changing clima ic condi-
ions (such as humidi y and empe a u e) in o accoun , k
[-] is a ac o desc ibing he basic esis ance o he con-
c e e mix u e o he ca bona ion p ocess. When designing
new s uc u es, he pa ame e s W and k can be de e -
mined based on da a measu ed on simila s uc u es in
simila clima ic condi ions; when e i ying exis ing s uc-
u es, hey mus be de e mined om measu emen s. Fo
modelling, he sophis ica ed model ecommended in he
ib Bulle in No. 34 [3] can be used, whe e he ca bona ion
dep h is de e mined based on he o mula:
𝑥𝑐(𝑡)=√2𝑘e𝑘c(𝑘 𝑅ACC,0
−1 +𝜀 )𝐶CO2∙𝑊(𝑡)∙√𝑡 (7)
The en i onmen al unc ion pa ame e , ke, includes he in-
luence o he mois u e con en o he conc e e su ace
(in luence o he ela i e humidi y RH [%]) on he alue o
he di usion coe icien , he pa ame e kc akes he in lu-
ence o he cu ing ime o he esh mix (pa ame e s c
[days] and bc [-]) on he e ec i e esis ance o he con-
c e e o ca bona ion in o accoun . The pa ame e 𝑅ACC,0
−1
[(m2/s)/(kg/m3)] is he in e se o he e ec i e esis ance
o he d y mix o ca bona ion, he alue o which is de e -
mined by es ing conc e e specimens exposed o condi-
ions in which he ca bona ion p ocess is accele a ed ( he
ACC es ). The pa ame e s k [-] and ε [(m2/s)/(kg/m3)]
co e he di e ences be ween samples es ed unde ac-
cele a ed condi ions and he s uc u e es ed unde na u-
al condi ions. The CCO2 pa ame e includes he e ec o
he CO2 concen a ion and he ime-dependen wea he
unc ion W( ) [-] accoun s o he wea he condi ions due
o ain (pa ame e s pSR [-], w [days] and bw [-]). We e e
he eade o [3] o de ailed ecommenda ions on he in-
di idual model pa ame e s.
In he case o chlo ide ions ing ess, he chlo ide concen-
a ion, C(x, ) [w . %/cemen ] ( ela i e o he weigh o
cemen ), a dep h x [mm] (mos o en a he dep h o he
conc e e co e o he s eel ein o cemen ) and ime
[yea s] can be exp essed o 1D cases by a simple analy -
ical o mula:
𝐶(𝑥,𝑡)=𝐶0+(𝐶S−𝐶0)[1−e (𝑥
2√𝐷c𝑡)] (8)
whe e CS [w . %/cemen ] is he chlo ide concen a ion on
he conc e e su ace, Dc [mm2/yea ] is he di usion coe -
icien o chlo ide pene a ion h ough he conc e e and C0
[w . %/cemen ] is he ini ial chlo ide concen a ion in he
conc e e. The solu ion o Eq. (8) is alid assuming homo-
geneous ma e ial ully sa u a ed wi h wa e and cons an
CS and Dc alues o e ime. Due o i s simplici y, his so-
lu ion has been used o se e al decades and is also ec-
ommended in a ious modi ica ions by a numbe o na-
ional and in e na ional no ma i e documen s. C i ical
poin s o he use o models o chlo ide pene a ion h ough
conc e e a e summa ized in e.g. [10], whe e, among o he
hings, he high a iabili y o modelling esul s caused by
o en conside able unce ain y in he alues o model pa-
ame e s is men ioned. Ano he widely discussed opic is
he simul aneous ac ion o mul iple deg ada ion p ocesses
and he ime dependence o di usion coe icien and su -
ace chlo ide concen a ion
The ime dependence o he di usion coe icien Dc can be
modelled acco ding o he ISO 16204 [5] using he o -
mula:
𝐷c(𝑡)=𝐷app(𝑡)=𝐷app(𝑡0)(𝑡0
𝑡)𝛼 (9)
whe e Dapp( 0) [mm2/yea ] is he ac ual alue o he di u-
sion coe icien de e mined a he e e ence ime, 0
[yea s] ( 0 = 28 days = 0.0767 yea s) and α [-] is he
ageing ac o o he conc e e, which akes he inc ease in
he esis ance o he conc e e o he pene a ion o ag-
g essi e subs ances due o i s ageing in o accoun , i.e., i
akes in o accoun he dec ease in he Dapp alue o e ime
due o he hyd a ion o he cemen componen s (changes
in he po e s uc u e). I should be no ed ha , bo h in he
design o new s uc u es and in he assessmen o he e-
sidual li e o exis ing s uc u es, he ageing ac o α should
be ob ained om obse a ions o s uc u es in si u, whe e
he conc e e composi ion, pe o mance and condi ions o
exposu e o agg essi e subs ances a e simila o hose o
he ac ual s uc u e. To calcula e he ageing ac o , obse -
a ions du ing a leas wo pe iods o exposu e (wi h a
su icien in e al be ween obse a ions) a e necessa y.
The model acco ding o he ib Bulle in No. 34 [3] accoun s
o he ime dependence o he di usion coe icien as ol-
lows:
𝐷c(𝑡)=𝐷app(𝑡)=𝑘en ∙𝐷RCM(𝑡0)∙𝑘u∙(𝑡0
𝑡)𝛼 (10)
whe e ken [-] is he en i onmen al pa ame e ha akes
he e ec o empe a u e on he di usion coe icien alue
(pa ame e s T [°C] and be [°C]) in o accoun , DRCM( 0)
[m2/s] is he chlo ide mig a ion coe icien , he alue o
which can be de e mined om he Rapid Chlo ide Mig a-
ion es (RCM), and k [-] is he con e sion ac o o he
di usion coe icien uni (ku = 3.1536∙106;
1 m2/s = 3.1536∙106 mm2/yea ). The concen a ion o
chlo ide ions is hen de e mined based on Eq. (8).
A he su ace o conc e e exposed o ex e nal chlo ide, a
laye whe e di usion is no he main p ocess and he chlo-
ide ions pene a ion p ocess di e s om he Fick's second
law o di usion due o exposu e o equen we ing and
subsequen e apo a ion, he so-called con ec ion zone,
Δx, usually o ms. Taking in o accoun his ac , he Eq. (8)
can be u he modi ied o he o m o [11]:
𝐶(𝑥,𝑡)=𝐶0+(𝐶S,∆𝑥−𝐶0)[1−e (𝑥−∆𝑥
2√𝐷app(𝑡)∙𝑡)] (11)
The maximum concen a ion o chlo ide ions he e is no a
he ou e su ace (x = 0), bu a ises p og essi ely a he
posi ion x = ∆x, wi h his hickness gi en in he ange o
6–11 mm. In he con ec ion zone, chlo ide concen a ions
can de ia e conside ably om no mal alues, so he model
acco ding o Eq. (11) neglec s hese alues and wo ks wi h
he so-called su oga e chlo ide concen a ion CS,∆x
[w . %/cemen ], which is applied only om a dep h
g ea e han he dep h o he con ec ion zone. This modi-
ica ion in de e mina ion o he chlo ide concen a ion in
ime is only meaning ul o long exposu e imes when he
con ec ion zone has o med and emained mo e o less
cons an o a long ime.
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To exp ess he ime dependence o he chlo ide concen-
a ion on he su ace o conc e e, CS, a linea o oo -
mean-squa e dependence on ime wi h cons an k acco d-
ing o [12] can be used:
𝐶S(𝑡)=𝐶0+𝑘𝑡 o 𝐶S(𝑡)=𝐶0+𝑘√𝑡 (12)
No e he e ha he unc ions in Eq. (12) a e no clea ly
sui able o desc ibing he chlo ide ions ing ess om he
sp eading sal s, which is ela ed o he change o seasons,
he ac ion o ain (washing he sal s om he conc e e su -
ace) and possibly o he clima ic o o he in luences. The
ime dependence o CS alue is also no included in ISO
16204 due o he complexi y o he whole issue and he e-
o e he use o hese models should always be ca e ully
conside ed in iew o possible e o s in he de e mina ion
o he se ice li e o s uc u es. As an example, he e ec
o seasonal applica ion o hawing sal s has been ad-
d essed o 2D cases by he cellula au oma a echnique,
see e.g. [13].
2.3 Simul aneous e ec s o en i onmen al ac-
ions and mechanical load
In ui i ely, i can be assumed ha deg ada ion p ocesses
will be as e in s uc u es wi h c acks. Howe e , nei he
he ib no he ISO commi ees ha e been able o come up
wi h any gene al model ha akes his e ec in o accoun .
The e o e, i was ag eed o use a simpli ied app oach ha
assumes ha ein o cemen co osion is no a ec ed up o
a ce ain c ack wid h. Depending on he se e i y o he
en i onmen al in luence and he sensi i i y o he s uc-
u e, he limi ing c ack wid h is usually gi en by a cha ac-
e is ic alue (i.e., 5% uppe quan ile) in he ange o 0.2
o 0.4 mm [2].
Acco ding o s udies epo ed in e.g. [14], he in luence o
s ess on he a e o ca bona ion and chlo ide di usion can
be easily aken in o accoun by means o co ec ion ac-
o s. The dep h o ca bona ion can be p edic ed wi h e-
spec o he s ess s a e acco ding o he o mula:
𝑥c(𝑡)=𝑘𝜎𝐴√𝑡 (13)
whe e he cons an A can be calcula ed using any sui able
ca bona ion model depending on he composi ion and cu -
ing o conc e e, ype o cemen , humidi y o he en i on-
men , CO2 con en , o o he pa ame e s (see e.g. he
abo e men ioned model acco ding o [3]). The co ec ion
ac o kσ [-] is de ined sepa a ely o elemen s unde en-
sion (s ess σ ) and comp ession (s ess σc) as:
𝑘𝜎(𝜎 /𝜎u, )=1+1.41(𝜎 /𝜎u, )+0.82(𝜎 /𝜎u, )2
𝑘𝜎(𝜎c/𝜎u,c)=1−2.27(𝜎c
𝜎u,c)+4.86(𝜎c/𝜎u,c)2 (14)
whe e σu, and σu,c ep esen he ul ima e ensile and com-
p essi e s ess o he conc e e. The dependence o kσ on
he ensile/comp essi e s ess a io and i s limi ing alue
is shown in Fig. 2. I is e iden ha a modes comp essi e
load, i.e., alues o (σc/σu,c) in he ange o 0 o app oxi-
ma ely 0.5, dec eases he a e o conc e e ca bona ion.
This e ec is due o pa ial closing o mic o-c acks unde
an applied comp essi e s ess. Howe e , i he load is in-
c eased abo e app oxima ely 50 % o he ul ima e com-
p essi e s eng h, he on-going closing mic o-c acks is
o e compensa ed by o ma ion o new c acks h ough
which ai CO2 can pene a e he conc e e. This accele a es
he ca bona ion p ocess. As he ensile load inc eases, he
ca bona ion p ocess is only accele a ed, since e en a mod-
es ensile s ess leads o he o ma ion and opening o
mic o-c acks, which se e as new pa hways o he pene-
a ion and mig a ion o CO2 in o he conc e e.
Figu e 2 Dependence o he kσ ac o on he a io o ensile/comp es-
si e s ess and i s limi alues
As in he case o ca bona ion, he models o chlo ide ions
ing ess can be ex ended by he e ec o mechanical load
on he s uc u e. Wi h espec o c acks in he conc e e,
o ma ed due o s ess om loading, he di usion o chlo-
ide ions is accele a ed. When he e ec o c acks is con-
side ed, he alue o he di usion coe icien (gene ally
deno ed by D) changes. This can be di ided in o wo pa s,
D0 [m2/s] and D [m2/s], as shown in Fig. 3. The di usion
coe icien D [m2/s] is de e mined wi h espec o he
wid h o he c acks o med, w [mm], and hei maximum
dis ance s ,max [mm] as [15]:
𝐷=(1− 𝑤
𝑠 ,max)𝐷0+𝑤
𝑠 ,max𝐷 (15)
whe e D0 is he alue o he di usion coe icien o in ac
conc e e and D is he alue o he di usion coe icien in-
side he c ack, he pa ame e s w and s ,max can be ob ained
e.g. by measu emen on a eal s uc u e o by calcula ion
acco ding o e.g. [16]. The alues o D can be de e mined
wi h espec o he c ack wid h as ollows [17]:
𝐷 ={0 m2/s
(0,16𝑤−3)∙10−10 m2/s
13∙10−10 m2/s (16a)
o he c ack wid h o :
{𝑤<30 μm
30 μm ≤ 𝑤≤100 μm
𝑤>100 μm (16b)
Figu e 3 Di usion coe icien D o chlo ide pene a ion h ough
c acked conc e e
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The applica ion o hese ela ions o modelling he simul-
aneous e ec o mechanical and en i onmen al load and
he e i ica ion o he abili y o hese models o achie e
esul s close o eali y is p esen ed in [18].
3 S udy on ma hema ical models wi h o wi h-
ou he e ec o mechanical load
A simple ein o ced conc e e ame b idge was selec ed o
he esidual se ice li e s udy. The b idge was pu in o op-
e a ion in 2019, and a e wo yea s o use (2021), a con-
inuous c ack g ea e han 0.1 mm wide was obse ed in
he middle o he span o he ame gi de (see Fig. 4).
Figu e 4 Con inuous c ack g ea e han 0.1 mm wide in he middle o
he ame span, eco ded a e wo yea s o b idge use – ups eam side
(le ), subs uc u e bo om iew (middle) and downs eam side ( igh )
The se ice li e o he b idge was analysed using ma he-
ma ical modelling o deg ada ion p ocesses. The inpu ma-
e ial pa ame e s conside ed we e as ollows: C30/37 con-
c e e class (exposu e class XD1 – medium we , humid
en i onmen and conc e e su ace exposed o chlo ides
dispe sed in ai ; wa e o cemen a io w/c = 0.55),
B500B,A ein o cemen p o ile o 32 mm diame e , con-
c e e co e 50 mm, s i ups o 12 mm diame e (i.e., main
ein o cemen co e a = 62 mm).
Fo modelling o deg ada ion p ocesses, he a o emen-
ioned models acco ding o he ib Bulle in No. 34 [3] and
ISO 16204 [5], espec i ely, we e used. The inpu pa am-
e e s we e de ined acco ding o Tab. 1. No e he e ha a
combina ion o de e minis ic and s ochas ic models was
used o de e mine he esidual se ice li e. De e minis ic
alues o ca bona ion dep h, chlo ide concen a ion and
loss o ein o cemen a ea o e ime we e calcula ed based
on he mean alues o he inpu pa ame e s (column
“Mean” in Tab. 1). To ake he a iabili y in ma e ial p op-
e ies and en i onmen al cha ac e is ics in o accoun , he
same ou pu s we e calcula ed using s ochas ic modelling
based on a sui ably chosen s a is ical model o he inpu
a iables (columns “COV” and “PDF” in Tab. 1, meaning
he coe icien o a ia ion and p obabili y densi y unc ion,
espec i ely).
The esul s a e summa ized in Fig. 5, which shows he
mean alues (de e minis ic model) ± s anda d de ia ions
(s ochas ic model) o ca bona ion dep h and chlo ide con-
cen a ion o e ime. FReET-D so wa e was used o s o-
chas ic modelling (see e.g. [19]), s a is ical cha ac e is ics
we e calcula ed based on one hund ed o andom simula-
ions.
No he de e minis ic no he s ochas ic calcula ion does
no assume ha he ein o cemen will be depassi a ed
due o ca bona ion o chlo ide a ack o e he design li e
o he s uc u e (solid lines in Fig. 5). The ein o cemen
depassi a ion is no assumed e en i he e ec o c acks
in conc e e is aken in o accoun (dashed lines in Fig. 5).
The esul s o he s ochas ic modelling hen show some
a iabili y in he modelling esul s. Howe e , using he ib
and ISO model, he dep h o conc e e co e and he c i ical
alue o chlo ide ions concen a ion a he dep h o he
conc e e co e is no eached e en when conside ing he
a iabili y o he inpu a iable, and i is he e o e clea
ha he design o he s uc u e wi h espec o he limi
s a e o du abili y is pe ec ly ine wi h espec o a su i-
cien conc e e co e .
Table 1 De ini ion o inpu pa ame e s
Inpu pa ame e [Uni ]
Mean
COV [-]
PDF*
No e
CCO2 [mg/m3]
820
0.12
N
RH [%]
70
0.07
B
Limi s: 0–100
c [days]
7
-
De .
bc [-]
–0.567
0.04
N
𝑅ACC,0
−1 [(m2/s)/(kg/m3)]
9.8×10–11
0.42
N
Limi s: 1×10–12–1×10–9
k [-]
1.25
0.28
N
ε [(m2/s)/(kg/m3)]
1×10–11
0.15
N
pSR [-]
1
-
De .
w [days]
60
-
De .
bw [-]
0.446
0.37
N
x [mm]
62
0.12
LN
σ /σu, [-]
1
-
De .
Assuming eaching he ensile
s eng h o conc e e
CS [w . %/cemen ]
0.465
0.75
N
Lowe limi : 0
C0 [w . %/cemen ]
0.0
-
De .
DRCM [m2/s]
1.97×10–11
0.20
N
be [-]
4526.85
0.15
N
T [°C]
10
0.05
N
α [-]
0.3
0.40
B
Limi s : 0–1
Cc [w . %/cemen ]
0.6
0.15
B
Limi s : 0.2–2.0
w [mm]
0.3
-
De .
Based on measu emen s (e -
ec o empe a u e changes
and conc e e sh inkage)
s ,max [mm]
600
-
De
Calcula ed acco ding o [16]
di [mm]
32
0.02
N
ico [-]
3
0.5
R
Rco [-]
6
-
De
Mean = 2 o uni o m co osion
No e: *De . = de e minis ic alue; B = be a, LN = logno mal, N = no mal, R = ec an-
gula p obabili y densi y unc ion
Al hough he ma hema ical modelling does no assume de-
passi a ion o he ein o cemen and subsequen co osion
o he ein o cemen ba s, a heo e ical analysis o he loss
o ein o cemen a ea was pe o med; see Fig. 6. He e,
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di e en o mulas we e used o ein o cemen a ea calcu-
la ions in o de o compa e he po en ial pe cen age loss
o ein o cemen a ea o e co osion p opaga ion pe iod.
Uni o m as well as pi ing co osion was compa ed o his
pu pose.
Figu e 5 Resul s o a s udy o he esidual se ice li e assessmen –
modelling o conc e e ca bona ion and chlo ide ions ing ess
Figu e 6 Resul s o a s udy o he esidual se ice li e assessmen –
modelling o ein o cemen co osion
The sa e y c i e ion was de e mined acco ding o he na-
ional s anda d ČSN 73 6221 [19] o e y poo (VI) o
eme gency condi ion (VII) as eaching he limi ing alue
o he co oded ein o cemen a ea a he le el o 15 %.
This alue would be achie ed a e app oxima ely 40 yea s
o p opaga ing co osion (Fig. 6). Based on he heo e ical
ma hema ical models a co osion loss o ein o cemen
a ea a he le el o 15 % was eached a e 40 yea s o
co osion p opaga ion wi h a p obabili y o 47.6 o 56.4 %
(acco ding o he o mula used o calcula e he esidual
a ea o co oded ein o cemen , A ). A no mal dis ibu ion
was assumed in his case. No e ha he p og ess o he
ein o cemen a ea is e y simila o uni o m and pi ing
co osion du ing he i s 50 yea s o p opaga ion. In he
ollowing yea s, pi ing co osion p og esses much as e .
I no addi ional measu es a e aken o ake in o accoun
he iden i ied de ec s in he o m o a localized c ack, he
s uc u e will be classi ied as being in s uc u al condi ion
VI o VII a e 40 yea s o use wi h his p obabili y. The e-
a e , es ic i e measu es will ha e o be aken wi hin he
amewo k o i s use, in he ex eme case esul ing in i s
closu e. I should be no ed he e ha i he loss o he e-
in o cemen a ea due o co osion up o 1 % is de ec ed,
i is ecommended o ecalcula e he load-bea ing capaci y
in ela ion o he classi ica ion o he s uc u e in s uc u al
condi ion IV (sa is ac o y). Based on ma hema ical mod-
elling, such a loss o ein o cemen a ea is achie ed wi h
a 50% p obabili y in 2 yea s o 11 yea s a e he ini ia ion
o co osion p ocess.
4 Conclusions
When he b idge owne /manage o s uc u al enginee is
aced wi h he need o assess he esidual se ice li e o
conc e e s uc u es in ela ion o ongoing deg ada ion p o-
cesses, ein o cemen co osion and he cu en s uc u al
condi ion, he use o app op ia e ma hema ical models in
combina ion wi h s ochas ic me hods appea s o be a e y
as and e icien ool compa ed o esidual se ice li e as-
sessmen p ocedu es based on knowledge o he se ice
li e o a simila s uc u e loca ed in simila condi ions o
based on accele a ed es ing. The use o ad anced me h-
ods o s ochas ic se ice li e analysis allows a g ea e in-
sigh in o he p og ess o he deg ada ion p ocesses aking
place, wi h he abili y o make p edic ions o he damage
o e ime. The alues o he heo e ical p obabili y o ail-
u e when limi ing alues a e exceeded, such as loss o e-
in o cemen a ea due o co osion, can also be calcula ed.
Finally, we no e ha ma hema ical and s ochas ic models
o conc e e deg ada ion and ein o cemen co osion e-
qui e knowledge o he inpu pa ame e s and hei p oba-
bilis ic models, which in some cases may appea o be a
disad an age. Howe e , he ma hema ical models can be
calib a ed based on he esul s o eal in-si u measu e-
men s and diagnos ic su eys ca ied ou . Sui able p oba-
bili y dis ibu ion unc ions and s a is ical cha ac e is ics
can also be ob ained om he a ailable li e a u e and
some no ma i e documen s.
5 Acknowledgemen s
The au ho s would like o g a e ully acknowledge he i-
nancial suppo o he Czech Science Founda ion p ojec
No. 22-00774S.
Re e ences
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