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HIERARCHICAL MODELLING OF UNCERTAINTY IN NDT TESTS
OF HISTORIC STEEL BRIDGES
Mi osla SÝKORA1, Jan MLČOCH1, Pa el RYJÁČEK2
1Depa men o S uc u al Reliabili y, Klokne Ins i u e, CTU in P ague, Šolíno a 7, P ague, Czech Republic
2Depa men o S eel and Timbe S uc u es, Facul y o Ci il Enginee ing, CTU in P ague,
Tháku o a 7, P ague, Czech Republic
[email p o ec ed], [email p o ec ed], pa el. yjace[email p o ec ed] u .cz
DOI: 10.35181/ ces-2020-0015
Abs ac . Sus ainable de elopmen can be suppo ed by
ex ending he se ice li es o exis ing oad and ailway
b idges. P ese a ion and upg ade should be based on
imp o ed su eys, moni o ing, eliabili y assessmen , and
s eng hening me hods. In he case o me allic ma e ials,
ha dness me hods (NDT) calib a ed by a ew ensile es s
(DT) we e shown o be associa ed wi h easonable
measu emen unce ain y. This con ibu ion discusses he
cu en p ac ice in assessmen based on NDT esul s and
in oduces he hie a chical modelling o he measu emen
unce ain y in ha dness es s. P elimina y esul s sugges
ha he a iabili y o ul ima e s eng h can ha dly be
es ima ed on he basis o NDTs only. I seems ha he
sys ema ic componen o measu emen unce ain y has a
lowe coe icien o a ia ion (3%) han he andom
componen (8%); he a iabili y o he la e may hus
o en exceed he a iabili y o he ul ima e s eng h o a
homogeneous ma e ial.
Keywo ds
Exis ing b idges, s uc u al assessmen , ha dness
me hods, hie a chical modelling, measu emen
unce ain y.
1. In oduc ion
The need o add ess sus ainabili y aspec s in cons uc ion
join ly wi h signi ican economic in e es s esul ed in
adding he assessmen and e o i ing o exis ing
s uc u es in o he e ision o Eu ocodes [1]. Unde his
highly p io i ised wo k i em, new Eu opean echnical
ules o assessmen we e de eloped [2] and a e in ended
o become pa o he p esen ly e ised EN 1990 o he
basis o design (p EN 1990-2).
Sus ainable de elopmen can be signi ican ly
suppo ed by using exis ing lines and c ossings, and his
leads o he u gen need o ex ension o se ice li es o
exis ing b idges [1]. P ese ing and upg ading o exis ing
b idges should be based on imp o ed su eys,
moni o ing, s uc u al assessmen , and s eng hening
me hods [3]. In he case o his o ic s eel (me al) b idges,
he conside able sca e o mechanical p ope ies and
missing design documen a ion necessi a e es s and
measu emen s o ob ain su icien in o ma ion o
s uc u al assessmen s [4] and [5]. The use o a ious
non- o mino -des uc i e es s (NDTs) is o en p e e ed
o des uc i e es s (DTs) o educe he cos o s uc u al
su ey and damage o he s uc u e.
Fo me allic ma e ials, ha dness me hods associa ed
wi h easonable measu emen unce ain y and may
p o ide a use ul basis o s uc u al assessmen s [6] and
[7]. To a oid g oss e o s in NDT esul s, s uc u e-
speci ic calib a ion o NDTs by a leas one ensile es ,
DT, is needed [8].
While he calib a ion based on a ew DTs educes he
sys ema ic componen o NDT measu emen unce ain y,
he andom componen (alea o y componen in he
modelling amewo k adop ed in he ollowing analysis)
canno be elimina ed. Taking a s a ing poin in he
p e ious s udies [6] and [7], his con ibu ion discusses
he cu en p ac ice in NDT assessmen , in oduces a
hie a chical modelling o he measu emen unce ain y in
ha dness es s, and quan i ies i s sys ema ic and andom
componen s.
2. Expe imen al Da abase
The da abase con ains 32 pai s o ul ima e s eng h alues
based on NDTs and DTs, aken om eigh his o ic
ailway b idges buil in he ea ly 20 h cen u y. The es
me hods unde in es iga ion a e as ollows:
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• DT esul s a e based on he ensile es acco ding o
ISO 6892 o ensile es ing o me als unde no mal
empe a u es. The es unce ain y is negligible
(coe icien o a ia ion, V < 1%) [9].
• NDT: he ha dness me hod acco ding o Leeb (see EN
ISO 16859, Pa s 1 o 3) conside ing an empi ical
ela ionship o con e ha dness alues o ul ima e
s eng h es ima es.
The da abase con ains es s on his o ic s eels (no w ough
i ons). The ma e ials a e assumed o p o ide a
homogeneous sample o he in es iga ion o
measu emen unce ain y.
3. Cu en P ac ice
When ma e ial s eng hs a e es ima ed om NDTs, i is a
common p ac ice o calib a e he mean o NDTs by a ew
DTs and o assume ha he s anda d de ia ion o NDTs,
σNDT, is a ep esen a i e (o conse a i e) es ima e o he
sca e o he ul ima e s eng h, σDT. Fig. 1 shows he
sample s anda d de ia ions [10] ob ained om NDT and
DT esul s o each o he eigh b idges in he da abase.
0
10
20
30
40
50
60
70
0 10203040506070
σ
NDT
[MPa]
σ
DT
[MPa]
Fig. 1: Sample s anda d de ia ions ob ained om NDT and DT esul s
o each o he eigh b idges in he da abase.
Fig. 1 shows ha σNDT unde es ima es σDT in hal o
he cases and p o ides a ough app oxima ion o σDT only.
I is emphasised ha his inding is only p elimina y – he
da abase is small, and he σNDT- and σDT- alues a e on
a e age es ima ed on he basis o me ely ou
measu emen s.
4. Hie a chical Model o
Measu emen Unce ain y
4.1. Sys ema ic and Random Componen s
In acco dance wi h he JCSS P obabilis ic Model
Code [11], i is assumed ha a NDT esul equals o he
DT ou come a ec ed by Θ an (componen o measu emen
unce ain y Θ, andom o each measu emen ) and by Θsys
(e o sys ema ic o a NDT su ey o a pa icula
s uc u e, bu andom amongs s uc u es).
Measu emen unce ain y depends on he combined
e ec o he imp ecision o he echnique, de ice and
hei applica ion. Based on he au ho s’ expe ience wi h
ha dness es s, he ac o s in luencing measu emen
unce ain y migh be classi ied as ollows:
- Dominan ly a ec ing Θ an:
• Be ween s uc u al membe s - s i ness and mass o
he specimen (a NDT should be applied in he s i and
hea y a eas, p e e ably s i ened by s i ene s o close o
hem; he es ing o hin pla es a om s i ene s mus be
a oided).
• Be ween s uc u al membe s – pa ly also he slope o
he in es iga ed membe (ho izon al s. e ical
measu emen s) – his unce ain y is commonly
compensa ed o elimina ed by mode n de ices.
• Homogenei y o ha dness o he ma e ial ( he ou e
pa s o he pla es ha e highe s eng h and ha dness han
he inne pa s due o he olling).
- A ec ing bo h Θsys and Θ an:
• Skills and expe ience o he wo ke
• Quali y o he specimen su ace ha mus be p ope ly
g inded o a smoo h su ace.
This s udy is ocused on he unce ain y in model
pa ame e s – p obabilis ic dis ibu ion pa ame e s o Θ.
Conside a ions o some aspec s (including he
epea abili y o a es ing de ice – p ope calib a ion o
he de ice, numbe o measu emen s a a pa icula
loca ion, o possible elimina ion o ex eme alues om
he sample o es ima e ha dness a a loca ion) a e beyond
he scope o he p esen ed analysis.
4.2. Model
Fo b idge i, a p obabilis ic ela ionship be ween NDT
measu emen s nd ij and DT measu emen s d ij aken a he
b idge can be es ablished:
θsys,i ~ LN(µsys, σsys), (1)
θ nd,ij ~ LN(µ nd, σ nd), (2)
nd ij = θsys,i θ nd,ij d ij ~ LN(θsys,i µ nd, θsys,i σ nd) d ij (3)
Using capi al le e s o deno e dis ibu ions o andom
a iables (also as in Sec ion 4.1) and lowe -case le e s o
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deno e pa icula ealiza ions o he dis ibu ions in
Eq. (1) - (3), he ollowing no a ion applies:
• Θsys – he same dis ibu ion o all b idges wi h θsys,i
as i s andom ealiza ion o b idge i;
• Θ an – dis ibu ion is iden ical o all NDTs and all
b idges wi h θ nd,ij being i s andom ealiza ion;
• LN – wo-pa ame e logno mal dis ibu ion wi h
mean μ and s anda d de ia ion σ;
• d ij – andom ealiza ion o he ma e ial p ope y ( he
ue alue – measu emen unce ain y in DTs is igno ed).
No e ha measu emen unce ain y is o en desc ibed by
a no mal dis ibu ion [12]. In he case unde
in es iga ion, he choice be ween hese wo ypes o
dis ibu ions is o low impo ance and a iabili y o he
measu emen unce ain y is low.
In Fig. 2, he DT o NDT a ios a e plo ed. The sho
ho izon al lines indica e he mean a io o a pa icula
b idge. In he p esen ed simpli ied app oach, each sho
line hus ep esen s a ealisa ion θsys,i and he sca e o
he do s a ound a espec i e line is indica i e o Θ an.
0.8
0.9
1.0
1.1
1.2
1.3
1.4
0 10203040
DT/NDT
measu emen no.
Fig. 2: DT o NDT a ios – illus a ion o sys ema ic and andom
componen s o he measu emen unce ain y ( he do ed lines
sepa a e he measu emen s o di e en b idges).
The eigh obse a ions o Θsys lead o he es ima e o he
mean µΘsys ≈ 1.03 and o a low coe icien o a ia ion
VΘsys ≈ 2.7%. Analysis o all 32 obse a ions e eals ha
he andom componen is unbiased and has a
compa a i ely highe coe icien o a ia ion, VΘ an ≈
7.8%. This inding is consis en wi h ha made in
Sec ion 3 – while he bias in NDTs can be co ec ed by
calib a ion conside ing DTs, he andom componen o
measu emen unce ain y is qui e signi ican and exceeds
he a iabili y o he ul ima e s eng h o a homogeneous
ma e ial in common cases.
Mo e e ined analysis o measu emen unce ain y,
based on he desi ed ex ension o he da abase, may
p o ide backg ound in o ma ion o in es iga ion o he
e iciency o calib a ion by DTs. Rela ed unce ain ies
can hen be quan i ied and conside ed in he amewo k
o he pa ial ac o me hod.
5. Discussion
Based on a limi ed da abase and using a simpli ied
app oach, his con ibu ion p o ides only he i s insigh
in o he hie a chical modelling o measu emen
unce ain y in ha dness es s. Fu he in es iga ions
should be ocused on he e ec o wi hin-s uc u e non-
homogenei y on measu emen unce ain y. I is widely
ecognized ha di e en s eng hs a e commonly
obse ed o olled sec ions and pla es as a esul o he
p oduc ion p ocess. The i s analysis sugges s ha a
sligh ly highe coe icien o a ia ion is ob ained o
pla es.
Rega ding p ac ical applica ions, i is emphasised ha
measu emen unce ain y can be conside ably educed,
mainly:
• Measu emen s should be conduc ed by an
expe ienced wo ke .
• Specimen su ace mus be adequa ely ea ed.
• Measu emen s should be aken a s i (and
possibly hea y) a eas, p e e ably s i ened by
s i ene s o close o hem; he es ing o hin
pla es a om s i ene s, a edges o pla es o a
loca ions whe e ib a ions and esonance may
occu mus be a oided.
Besides he desi ed ex ension o he da abase, u u e
esea ch should p o ide answe s o he ollowing
ques ions:
• Can di e en ha dness es me hods (s a ic o
dynamic) be desc ibed by he same model o
measu emen unce ain y? Fi s esul s seem o sugges so
[6] and [7].
• Is he mul iplica i e o ma o Θ—see Eq. (1) and
(2)—app op ia e, o should he addi i e o ma o hei
combina ion be p e e ed?
• Can he andom and sys ema ic componen s o
measu emen unce ain y be desc ibed by he same
dis ibu ions o a ious s uc u es?
• Wha is he s a is ical unce ain y in Θsys and Θ an
using he equen is o Bayesian app oach?
• Is he hie a chical modelling needed o p ac ical
applica ions o would i be su icien o desc ibe
measu emen unce ain y by a single andom a iable as
was conside ed e.g. in [6] and [7]?
I migh well appea ha he andom componen o
measu emen unce ain y is domina ing and i may be
su icien in p ac ical applica ions o desc ibe he
measu emen unce ain y igno ing he sys ema ic
componen .
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6. Conclusion
The limi ed da abase p o iding he basis o his s udy
makes i possible o p o ide only p elimina y concluding
ema ks abou measu emen unce ain y in ha dness
me hods o his o ic s eels:
• Va iabili y o ul ima e s eng h can ha dly be
es ima ed on he basis o NDTs only.
• Sys ema ic componen o measu emen unce ain y is
ound o ha e a signi ican ly lowe coe icien o
a ia ion (3%) han he andom componen (8%).
• The a iabili y o he la e may hus o en exceed he
a iabili y o he ul ima e s eng h o a homogeneous
ma e ial and he e iciency o calib a ion by DTs can be
doub ul. A mo e easonable app oach seems o be o
e i y homogenei y o he ma e ial by NDTs and
es ablish he model o ul ima e s eng h om DTs.
• The opics o u he esea ch include in es iga ions
in o he e ec o wi hin-s uc u e non-homogenei y on
measu emen unce ain y, unce ain ies in a ious NDT
me hods, and app op ia e app oaches o hie a chical
modelling and o s a is ical in e ence.
Acknowledgemen s
This wo k was suppo ed by he Minis y o Cul u e o
he Czech Republic unde G an DG18P02OVV033 ‘‘The
Me hods o Achie ing he Sus ainabili y o Indus ial
He i age S eel B idges”.
Re e ences
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2015. 125 pp. ISBN 1018-5593. DOI:
10.2788/095215.
[2] CEN/TC250/WG2. Assessmen o Exis ing
S uc u es (d a o he echnical speci ica ion, Ma
2019, CEN/TC 250 N 2176). CEN TC250/ WG2,
2019. 54 pp.
[3] BIEN, J., L. ELFGREN and J. OLOFSSON.
SUSTAINABLE BRIDGES. Assessmen o Fu u e
T a ic Demands and Longe Li es. W oclaw:
Dolnoslaskie Wydawnic wo Edukacyjne, 2007. 490
pp. ISBN 978-83-7125-161-0.
[4] MACHO, M., P. RYJACEK and J.C. MATOS.
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[5] RYJACEK, P. The diagnos ic echniques o he
assessmen o he his o ical s eel b idges. In IABSE
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[6] SYKORA, M., J. MLCOCH and P. RYJACEK.
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S eels Using Non-Des uc i e Techniques.
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Assessmen o exis ing s uc u es. Gene e,
Swi ze land: ISO TC98/SC2, 2010. 44 pp.
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Abou Au ho s
Mi osla SÝKORA was bo n in České Budějo ice,
Czech Republic. He ecei ed his M.Sc. in 2001 and Ph.D.
in 2005 om Facul y o Ci il Enginee ing, CTU in
P ague. His esea ch in e es s include basis o s uc u al
design, 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.
Jan MLČOCH was bo n in Mladá Bolesla , Czech
Republic. He ecei ed his M.Sc. in 2015 om Facul y o
Ci il Enginee ing, CTU in P ague. His esea ch in e es s
include unce ain y quan i ica ion and p obabilis ic
eliabili y analysis o ein o ced conc e e s uc u es.
Pa el RYJÁČEK was bo n in Plzeň, Czech Republic.
He ecei ed his M.Sc. in 2000 and Ph.D. in 2003 om
Facul y o Ci il Enginee ing, CTU in P ague. His
esea ch in e es s include s eel b idges (connec ions,
s abili y, composi e ac ions), FRP applica ions in b idge
enginee ing, b idge assessmen and upg ading, a igue
assessmen , and ack-b idge in e ac ion.