SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 |NUMBER: 2 |2019 |DECEMBER
Legion B idge in P ague – Assessmen o S one
A ches
Ma ek VOKÁL, Michal DRAHORÁD
Depa men o Conc e e and Mason y S uc u es, Facul y o Ci il Enginee ing,
Czech Technical Uni e si y in P ague, Thaku o a 7/2077, 166 29 P ague 6, Czech Republic.
ma ek. ok[email p o ec ed], mic[email p o ec ed]
DOI: 10.35181/ ces-2019-0023
Abs ac . The a icle deals wi h a ailable enginee ing
assessmen me hods o he s one aul b idges acco d-
ing o ele an s anda ds. The a icle summa izes a i-
able me hodologies used o design and assessmen o
mason y aul s in las pe iod. All me hods we e ap-
plied o assessmen o he Legion B idge o e Vl a a
Ri e in P ague and esul s we e compa ed. The mod-
els we e alida ed by esul s o s a ic load es .
Keywo ds
Mason y a ch b idge, Vaul , B idge, Assess-
men , Load ca ying capaci y, Th us line
1. In oduc ion
When assessing exis ing b idges, he equi emen o
su icien mechanical esis ance and s abili y is usually
gi en by he maximum load ha he s uc u e is able o
ca y sa ely - he maximum weigh o he oad ehicle
which can pass he b idge unde he speci ied condi-
ions, i.e. he load ca ying capaci y.
I is necessa y o know he ac ual ma e ial cha ac e -
is ics o he ca ying elemen s o he s uc u e and he
geome y o he s uc u e as a basis o de e mining he
load-ca ying capaci y. Deg ada ion o he s uc u e in
he cou se o ime changes bo h he ma e ial and geo-
me ic pa ame e s o he ca ying and non-ca ying el-
emen s o he b idge. The e o e, inding ou he deg ee
o deg ada ion is c i ical o de e mine he load-ca ying
capaci y. De e mining ac ual de ec s o he b idge, he
deg ee o deg ada ion and he eal b idge geome y a e
objec i e o diagnos ics.
The load ca ying capaci y is in luenced by he ac
ha he b idge s uc u es we e designed o be loaded
acco ding o he s anda ds alid a he ime o b idge
cons uc ion. The equi emen s o load o be ans-
e ed by he b idges a e inc easing. The Legion B idge
was designed o be loaded by a he d o ca le, and a
load es o a ca le he d was also ca ied ou on he
b idge o Empe o F ancis, which is name o he Legion
B idge a he ime o comple ion. The e o e, a numbe
o his o ic s uc u es do no ha e he load ca ying ca-
paci y ha he adminis a o o b idge wan s (e en in
s a e wi hou de ec s and deg ada ion).
De e mina ion o he load bea ing capaci y o he
exis ing b idge oday is go e ned by he same p inciples
as he design o new s uc u e (see applicable echnical
s anda ds and egula ions EN, DIN, and MVL).
In case o mason y aul s uc u es his ask is com-
plica ed by he s uc u al beha iou and ypical p op-
e y o he ma e ial - a negligible ensile s eng h, see
[1] and [9]. Due o hese ac s, he p ocedu es o de e -
mina ion o he load ca ying capaci y a e non-linea
and mus in ol e a la ge numbe o pa ame e s wi h
signi ican a iabili y. The e o e, i is e y di icul o
se up a simple analy ical model and special p og ams
de eloped di ec ly o aul s uc u es a e usually used.
In case o oad b idges h ee kinds o load ca ying
capaci y (acco ding o [6]) can be calcula ed :
•Vn - no mal load ca ying capaci y o he b idge
– ep esen s he maximum weigh o one ypi-
cal lo y, which may pass he b idge wi hou any
es ic ions (posi ion limi ed by sa e y ba ie s
only).
•V - exclusi e load ca ying capaci y – ep esen s
he maximum weigh o one uck, which can pass
he b idge as single ehicle in any posi ion o lane
espec i ely (no o he a ic loads excep pedes-
ians is pe mi ed). I can mo e anywhe e on he
b idge.
c
2019 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 71
SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 |NUMBER: 2 |2019 |DECEMBER
•Ve - excep ional load ca ying capaci y - which
ep esen s he maximum weigh o special ehi-
cle, which can pass he b idge unde special con-
di ions (speci ied eloci y, speci ied pa h and ec-
cen ici y).
De e mina ion o each load ca ying capaci y
p esen s sepa a e calcula ion in bo h limi s a es (SLS
and ULS) conside ing all o he ele an loads ( empe -
a u e, wind, lood e c.).
The calcula ion o load ca ying capaci y was ca ied
ou using he esul s o diagnos ics acco ding o chap-
e 3. The condi ions o eliabili y used o e i y he
s uc u e see in chap e 4. The me hods o de e min-
ing he in e nal o ces and s esses see in chap e 5.
2. B ie his o y and
desc ip ion o he b idge
The Legion b idge connec s he Old own wi h he
Lesse own o P ague h ough he S řelecký Island.
The o egoe b idge o Legion B idge was B idge o
Empe o F an išek and i was second b idge in P ague
inished in 1841. The Legion B idge was cons uc ed a
he posi ion o B idge o Empe o F an išek. The con-
s uc ion o new b idge an om 1898 o 1901. In con-
as o he o iginal b idge, he supe s uc u e is made
o a massi e s one aul s uc u e. The cons uc ion o
he b idge consis s o nine la aul s o di e en spans:
26.6+34.3+38.5+42.0+27.8+27.8+31.9+28.7+25.6m.
Two aul s abo e he S řelecký Island a e aul s o ci -
cula segmen s, he o he aul s a e ellip ical. The con-
s uc ion o he b idge is made o g ani e blocks wi h a
gap o 12−15 mm illed wi h cemen mo a . The s one
acade o on walls o ligh sands one and ed g an-
i e symbolizing na ional colo s. The b idge has wide
oo ways and mo o way lanes, elec ic acks we e also
buil on he b idge, ams un along an exis ing b idge
om June 17, 1901.
The Legion B idge is an immo able cul u al monu-
men and is he e o e p o ec ed acco ding o he p o-
isions o law On S a e Monumen Ca e, as amended.
Due o he ac ha i is a building loca ed in he
e i o y o he P ague His o ical Rese e (PPR), he
p o isions o he Go e nmen Dec ee On he His o ical
He i age in he Capi al Ci y o P ague. The his o -
ical monumen in P ague, ep esen ing he his o ical
cen e o P ague, was included in he UNESCO Wo ld
He i age Lis in 1992.
3. Diagnos ics o he b idge
In o de o ind he geome ic and ma e ial p ope ies
o he b idge, ollowing p ocedu es we e ca ied ou .
Fo de ailed esul s see [11]:
1. de ailed mapping o all isible damages,
2. geode ic su ey,
3. geo echnical bo eholes,
4. co e bo eholes om he supe s uc u e (ge ing
E, b, m),
5. geo ada su ey o hickness o a ches,
6. con inuous measu emen o empe a u es,
7. di ing su ey o subs uc u e,
8. s a ic and dynamic load es – see chap e 7.5.
3.1. De ailed mapping
As an example o de ailed mapping o isible dam-
ages, which was done oge he wi h he acous ic es
o deg ada ion o s ones o a ches, ollowing igu e is
shown:
Fig. 1: Mapping o isible damages in he spand el wall nea
pie No. 2
Because o he c acks in he spand el walls, he wall
is conside ed jus as a load in he models, no as a
ca ying elemen .
c
2019 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 72
SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 |NUMBER: 2 |2019 |DECEMBER
3.2. Geo echnical bo eholes
Pie s numbe 2, 3, 4, 6, 8 we e d illed om he b idge
deck o he ounda ions. F om ha , we ge he in-
o ma ion abou hickness and ma e ial p ope ies o
asphal laye s, back ill, mason y o pie s and ounda-
ion g ound. I was ound ou , ha mason y o pie
has much lowe s eng h, han he supe s uc u e, see
chap e 3.3.
3.3. Co e bo eholes
Comp essi e s eng h was es ed on 34 g ani e spec-
imens om supe s uc u e. The measu ed a e age
s eng h was b= 132 MPa. Mo a o supe s uc-
u e has a e age s eng h 25.8MPa, es ed on 114
specimens.
Comp essi e s eng h was es ed on 131 g ani e and
sedimen a y ock specimens om subs uc u e. The
measu ed a e age s eng h was m= 86 MP a. Mo -
a o supe s uc u e has a e age s eng h 25.2MPa,
es ed on 169 specimens.
The cha ac e is ic comp essi e s eng h o mason y
k=K α
b β
m,(1)
was calcula ed 28.9MPa o supe s uc u e and
22.8MPa o subs uc u e (which a e e y high al-
ues),
whe e:
Kis coe icien depending on ype o mason y and
mason y elemen s,
αis coe icien depending on ype join s and mo a ,
βis coe icien depending on ype o mo a .
The Young modulus o g ani e o supe s uc u e was
ound 41.3GPa on 17 specimens, Young modulus o
g ani e o subs uc u e was ound 31.1GPa on 29 spec-
imens.
3.4. Geo ada su ey o hickness o
a ches
The Geo ada me hod (GPR) is based on he p inciple
o ansmi ing high- equency elec omagne ic wa es
in o he examined en i onmen and hen egis e ing
he wa e image o e lec ed wa es. The wa e image is
a ec ed by local inhomogenei ies, especially wi h di -
e en conduc i i y and o he elec omagne ic p ope -
ies. Inhomogenei ies can ha e bo h plana cha ac e
(discon inui y su aces, s uc u al in e aces) and lo-
cal cha ac e (eg, ca i ies, e c.). Inhomogenei ies a e
mani es ed by ampli ica ion o he egis e ed e lec ed
signal ampli ude. In Figu e 2 he e a e blue lines ep-
esen ing he a ch shape aken om geode ic su ey.
The yellow a ea and whi e lines a e aken om he
Geo ada me hod.
Fig. 2: Geo ada me hod in span No. 9
Measu ing in longi udinal di ec ion o he b idge
showed good ag eemen wi h he a chi e documen a-
ion. On he aul o span 7 was ound 0.15 −0.2m
conc e e laye , which was no expec ed. Measu ing in
ans e se di ec ion showed cons an hickness, which
was expec ed.
4. Limi s a es
4.1. Ul ima e limi s a e
In he ul ima e limi s a e (ULS) he beha iou o
he s uc u e jus be o e he collapse is in es iga ed.
Fo he bea ing capaci y de e mina ion load ac o s
acco ding o app op ia e EN a e conside ed (1.35 o
dead load and 1.35 o li e load). Gene ally, i is as-
sumed ha plas ic hinges a e ully de eloped h ough
he s uc u e. Fo de ails on mason y a ch b idges be-
ha iou see in [1] and [2]. Resis ances o axial and
shea o ces a he ULS can be w i en acco ding o [3]
as:
NRd = db(h−2eu),(2)
VRd = ( k0+ 0.4σd)b(h−2eu)/γM,(3)
whe e: dis he design s eng h o mason y in com-
p ession, k0is he cha ac e is ic alue o ini ial shea
s eng h a no mal s ess equal o 0, b, h is wid h o
heigh - espec i ely, euis he eccen ici y o he esul-
an axial o ce in c oss-sec ion a he ul ima e limi
s a e, σdis he design comp essi e s ess in he com-
p essed a ea a he ul ima e limi s a e ( he s ess is
uni o mly dis ibu ed, see Figu e 3), 0.4is he coe i-
cien o ic ion in he mason y join , γMis he ac o
o he ma e ial.
c
2019 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 73
SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 |NUMBER: 2 |2019 |DECEMBER
Fig. 3: S ess dis ibu ion o mason y c oss-sec ion a he ul i-
ma e limi s a e.
4.2. Se iceabili y limi s a e
The se iceabili y limi s a e (SLS) desc ibes he be-
ha iou o he s uc u e unde o dina y ope a ing con-
di ions. Ful illing he condi ions o se iceabili y limi
s a e p o ides he equi ed p ope ies and beha iou o
he s uc u e h oughou i s li e ime. In e ms o se -
iceabili y limi s a e, c ack wid h and s uc u al s ess
unde ope a ing load a e e i ied (load ac o equals o
1.0). In e ms o e i ica ion o aul s uc u es, i is
necessa y o e i y he maximum axial s ess in he
c oss-sec ion and he heigh o he comp essed a ea a
he c oss-sec ion (see [4] and [3]). Elas ic beha iou
o he s uc u e is conside ed wi h a linea dis ibu-
ion o axial s ess in he comp essed a ea o he c oss
sec ion. Tensioned pa o he sec ion is excluded o
s ess de e mina ion (see Figu e 4 ):
Fig. 4: S ess dis ibu ion o mason y c oss-sec ion a he se -
iceabili y limi s a e.
σn,max =NEk
3b(h−2e)≤0.45 k,(4)
hc≥h
2−→ e≤h
3,(5)
e=MEk
NEk
,(6)
whe e: MEk, is cha ac e is ic momen caused by
load, NEk is cha ac e is ic no mal o ce caused by load.
5. Me hods o calcula ion -
a ch
1. G aphical me hod (con ols all he equi emen s)
2. Linea calcula ion (con ols all he equi emen s)
(a) Beams – 2D o 3D
(b) Plane-s ess elemen s – 2D
(c) 3D solid elemen s
3. Non-linea calcula ion (con ols SLS equi e-
men s)
(a) 2D – plane
(b) 3D – solid elemen s – no used in his a icle
4. Equilib ium me hod — Limi S a e:Ring (con ols
only collapse o he s uc u e)
5.1. G aphical me hods
Va ious g aphical me hods had been used un il com-
pu e aided design came o enginee ing p ac ice. I
p o ides e y simple and quick design app oach inde-
penden on a ch b idge shape. G aphical me hods a e
based on he h us line de e mina ion. Th us o ce
a he c oss sec ion can be ound as cen oid o he
axial s ess diag am. When he h us line is known,
s ess a he c oss-sec ion can be calcula ed as well.
The me hod o inding he h us line uns in ollowing
o de (symme ic a ch acco ding o [5]):
1. Di ide he a ch in 2 symme ic pa s, ind he
weigh o one hal and om he geome y o a ch
we ind o ce H (ho izon al o ce in he op o a ch)
2. Di ide one hal o a ch in o se e al pa i ions ( e -
ical lines can be used o di ide)
3. D aw g aphical ep esen a ion o all pa s – Fi -
size o ec o in chosen scale, ac s in i s cen oid
4. Fo ce H we loca e o example in he uppe bound
o c oss sec ion co e and eac ion in he lowe
bound o c oss sec ion
5. Fo ge ing he esul ing o ce R1 in i s pa i ion,
we g aphically add he o ce Fi o H, o ge ing
nex esul ing o ces in each pa i ion, we g aphi-
cally add he o ces Fi o p e ious esul ing o ce
Basic p inciple o calcula ion is shown in Fig.5 and
[8].
c
2019 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 74
SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 |NUMBER: 2 |2019 |DECEMBER
Fig. 5: Basic p inciple o he g aphical me hod.
5.2. Linea calcula ion
S uc u al analysis using linea calcula ion me hod was
done in wo di e en op ions. In he i s one he a ch
was ep esen ed by sequence o beams in i s cen e
line. The back ill was ep esen ed by e ical beams
p o ided in app op ia e longi udinal dis ance, see Fig-
u e 6. While modelling using linea calcula ion, one
should no o ge , ha his calcula ion me hod doesn’
ake in o accoun ma e ial non-linea i y (and geome y
changes due o excluding he ensioned pa o c oss
sec ion). Fo he second op ion he a ch and back ill
we e modelled by he 3D solid elemen s wi h a ious
mechanical p ope ies – see Figu e 11.
Fig. 6: Linea 2D beam model.
5.3. Non-linea calcula ion
Fo pe o ming an non-linea analysis and assessmen
o he s uc u e, he ma e ial – mo a and mason y
elemen s - is homogenized o p ese e i s p ope ies
in ela ion o he eal beha iou o he s uc u e o
i s pa . I is assumed ha he dimensions ( hick) o
he mason y elemen s and join s be ween hem do no
signi ican ly a ec he dis ibu ion o s ess in he ma-
son y elemen . The eal s ess-s ain diag am o he
mason y shows non-linea beha iou (see [1]) pa icu-
la ly due o negligible ensile s eng h. In his a icle,
i is conside ed ha he ma e ial ac s only in comp es-
sion and when he ensile s ess occu s, c acks open up,
see Figu e 7). I , subsequen ly, (e.g. in ano he load
combina ion) he ensile s esses in he c oss sec ion
disappea , he c acks close and he c oss-sec ion ac s
again as ull.
Fig. 7: Non-linea beha iou o mason y.
The s uc u al model in p og am Midas was p e-
pa ed using plane-s ess elemen s, modelling he join s
be ween he g ani e blocks as se o elas ic links wi h
he p ope y "Comp ession only", see Figu e 8.
Fig. 8: Elas ic links be ween nodes in join s o mason y.
5.4. Equilib ium me hod on igid
blocks
Limi S a e:RING is a special analysis so wa e o
checking he load bea ing capaci y o he aul in he
plane o he longi udinal sec ion o he b idge s uc-
u e, including he load dis ibu ion by he back ill. I
uses equilib ium equa ions on he pa s o aul ac as
a igid bodies, he s uc u e is di ided in o igid bod-
ies depending on o ming o he plas ic hinges in loca-
ions wi h he lowes heigh o comp essed a ea. Load
dis ibu ion is conside ed acco ding o Bousinesq, see
example in Figu e 9. Fo mo e de ails see [7].
c
2019 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 75

SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 |NUMBER: 2 |2019 |DECEMBER
Fig. 9: Ve ical a ic load dis ibu ion (dispe sion) o he aul
conside ed o a ch modelling in Limi S a e:RING so -
wa e.
6. Me hods o calcula ion -
ans e se di ec ion
6.1. Linea calcula ion
1) Beam elemen s
Modelling using his me hod is done by ep esen ing
he a ch by beams in i s middle line and di iding he
back ill in chosen in e al o ep esen i by beams as
well, see Figu e 6. I he model shown in Figu e 6
is copied se e al imes in he ans e se di ec ion he
analysis model ep esen ing he a ch as a body can be
a anged. The s i ness o ans e se beams is chosen
as a s i ness o a ch. Such 3D linea model can be
used o s udy he e ec o eccen ici y o li e load on
he bending momen s dis ibu ion in he ans e se di-
ec ion – see Figu e 10.
Fig. 10: Linea 3D beam model.
2) Solid elemen s
Ano he op ion o linea analysis is o model he body
o he s uc u e by he 3D solid elemen s – see Figu e
11. Bo h longi udinal and ans e se di ec ion can be
modelled by his way.
Fig. 11: 3D solid model.
6.2. E ec i e wid h
P inciples o "modelling" o he b idge span 4 in ans-
e se di ec ion using e ec i e wid h can be seen om
Figu e 12. This way o modelling is used in mos codes.
I conside s conse a i e idea, ha non-loaded lane o
a ch doesn’ ca y any load. So he shea and bend-
ing s i ness be ween loaded and non-loaded elemen s
is conside ed equal ze o.
Fig. 12: Calcula ion o e ec i e wid h.
7. Resul s
7.1. G aphical me hod
Resul o g aphical me hod o span 4 a e shown in
Figu e 13.
Fig. 13: G aphical solu ion o he span 4 om he a chi e doc-
umen a ion.
c
2019 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 76
SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 |NUMBER: 2 |2019 |DECEMBER
7.2. Linea calcula ion
The bending momen s on he beams a e shown in Fig-
u e 14.
Fig. 14: Bending momen on he beams ep esen ing a ch in
longi udinal di ec ion.
Fig. 15: Resul ing s ess om he load o exclusi e load model.
7.3. Non-linea calcula ion
P incipal s ess in he a ch o span 4 can be seen in
Figu e 16.
The legend o cu es in he Figu e 17 is ollowing:
•sw means sel weigh ,
•Nmeans non-linea combina ion,
•oh means hea -up,
•och means cooling,
•4V nT!means load by li e load.
Fig. 17: No mal s ess dis ibu ion in he middle o span 4 e -
sus he c oss sec ion heigh .
7.4. Shea be ween he blocks
As a demons a ions o ailu e mode, pic u e o span 4
om p og am Limi S e:RING is a ached:
Fig. 18: Failu e mode o span 4
The esul ing shea o ce ac ing on he c oss sec-
ion depends on he geome y o he a ch. In beam
model, i depends on chosen geome y o axis o he
beams. In solid model, i depends on choice o ma e-
ial model – linea o non-linea . In Fig. 20 can be
seen, ha changes in geome y acco ding o ma e ial
non-linea i y a ec s esul ing shea o ce highly. Fig-
u e shows esul ing shea o ce om he linea beam
model wi h changed axis nea he bo om o a ch.
7.5. S a ic load es
Six ehicles o weigh 31.6,31.2,31.7,31.5,31.7,31.5
we e used o he s a ic load es . Tes was ca ied ou
acco ding o [10] in spans 3, 4, 5, 6. In his a icle, only
esul s o span 4 a e p esen ed. The measu ed displace-
men s a e qui e small in compa ison o p eciseness o
measu emen , which is equal 0.25 mm. Fou unc ions
o ans e se e ical displacemen a e compa ed in he
Figu e 21:
1. linea beam model,
2. linea solid model,
3. "non-linea beam model",
4. "non-linea solid model".
c
2019 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 77
SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 |NUMBER: 2 |2019 |DECEMBER
Fig. 16: P incipal s ess om he non-linea model.
Fig. 19: Changes o geome y o beam axis depending on ec-
cen ici y o load
Resul s om he e ec i e wid h calcula ion is no
p esen ed in he plo , because i is e iden , ha i
is e y conse a i e – esul ing displacemen on he
le and igh edges would be ze o, which is no
ue. The unc ion "non-linea beam model" and "non-
linea solid model" was c ea ed by mul iplying he lin-
ea esul by knon, which was calcula ed as knon =
wnon−linea /wlinea , whe e wnon−linea is displacemen
om he 2D non-linea model loaded by sel weigh
and wlinea is displacemen om he linea solid model
loaded by sel weigh .
The esul ing "non-linea " calcula ed displacemen s
a e in he ange o measu ed displacemen s +- p ecise-
ness (which is equal o 0.25 mm). The esul s app o e
he non-linea beha iou o he b idge.
Fig. 20: Resul ing shea o ce depending on eccen ici y o load
Fig. 21: Displacemen compa ison
8. Compa ison o me hods and
discussion
8.1. Sel weigh – he main load
In he Figu e 22 we can see he eccen ici y om he
linea , non-linea model and om g aphical solu ion
om he a chi e documen a ion. I is e iden , ha
c
2019 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 78
SECTION BUILDING STRUCTURES & STRUCTURAL MECHANICS VOLUME: 19 |NUMBER: 2 |2019 |DECEMBER
eccen ici y om he non-linea model is highe , as ex-
pec ed, because ension, which is allowed in he linea
model pushes he esul an h us line o he cen oid
o he c oss sec ion. The non-linea analysis is mo e
ime consuming (on he e o o he enginee as well
as he e o o he compu e ), bu he eal beha iou
o he s uc u e is be e desc ibed by he non-linea
model (see he beha iou o mason y in Figu e 7 and
21). The esul s om he non-linea model a e mo e
dange ous and a e close o he limi s a es.
Fig. 22: Compa ison o linea , non-linea and g aphical me hod
o span 4.
8.2. T a ic load and i s dis ibu ion
in ans e se di ec ion
Load dis ibu ion in he ans e se di ec ion was com-
pa ed on h ee models – 3D solid, 3D beam model and
e ec i e wid h model. Resul can be ound in Figu es
14, 15 and 12. 3D linea solid model assumes linea be-
ha iou in all di ec ions, he e o e gi es he mos non-
conse a i e esul s. The mos conse a i e model is
e ec i e wid h model, because i en i ely excludes pa
o c oss sec ion. The eal beha iou is somewhe e be-
ween. The beam model gi es esul s be ween he wo
men ioned me hod, in opinion au ho is he e o e he
mos eal. The beam model is much simple and he
s i ness in ans e se di ec ion can be easily changed.
Real 3D solid model (which conside s non-linea be-
ha iou ) is complica ed wi h he ac , ha we don’
know many c ucial cha ac e is ics o mason y – such
as bending and shea s i ness o mo a be ween he
blocks – nei he in longi udinal no he ans e se di-
ec ion. Tha is he eason he 3D solid model is ec-
ommended jus o special s uc u es and i we know
he pa ame e s.
8.3. Final esul s o load ca ying
capaci y – no mal s ess
The inal esul s o load ca ying capaci y e lec s bo h
he esul s o modelling he a ch i sel and modelling
o ans e se di ec ion.
Tab. 1: Resul ing load ca ying capaci y – no mal s ess. (m.
means model)
A ch m. T ans e se m. Vn V Ve
Linea 3D beam 41 122 230
Non-linea e ec i e wid h 32 83 185
RING e ec i e wid h 46 105 182
I is in gene al known, ha Limi S a e:RING gi es
non-conse a i e esul s. Non-linea model gi es con-
se a i e esul s, i s because o he me hod o assess-
ing he load dis ibu ion in ans e se di ec ion, sec-
ond because o modelling o he a ch i sel . The lin-
ea model esul s a e be ween wo men ioned me hod.
Fo Ve and V , he beam model gi es he mos non-
conse a i e esul s. The load is concen a ed o small
s ip o he a ch in e ec i e wid h model, bu in 3D
beam model all he beams ca y pa o he load.
8.4. Final esul s modelling – shea
s ess
When calcula ing he esis ance o mason y blocks o
he shea , c ucial is ic ion coe icien . In Limi -
S a e:RING, de aul alue 0.6 is conside ed (acco ding
o labo a o y es s, see [12]). Fo compa ison, alues
om 0.597 o 0.705 we e ob ained acco ding o [13].
Acco ding o [3], coe icien 0.4 should be used. Fo
he compa ison, he coe icien 0.4 is used o all mod-
els.
Tab. 2: Resul ing load ca ying capaci y – shea s ess. (m.
means model)
A ch m. T ans e se m. Vn V Ve
Linea 3D beam 0 0 0
Non-linea e ec i e wid h 42 111 240
RING e ec i e wid h 35 79 161
As can be seen in Figu e 20 and 19, geome y
changes a e c ucial and model, ha doesn’ conside
ha , gi es un eal shea o ces. This is he case o lin-
ea beam models o aul s o low sagi a.
9. Conclusion
Se e al models o he Legion b idge we e ca ied ou .
Resul s we e alida ed by he s a ic load es . Mod-
elling o a ches showed, ha he mos eal beha iou
o a ches desc ibes he non-linea model, because i
conside s he non-linea i y, which impac s he calcu-
la ion he mos – negligible s eng h in ension. The
esul s om non-linea modelling a e non-conse a i e
in compa ison o o he me hods. Modelling o load dis-
ibu ion in ans e se di ec ion showed, ha 3D solid
c
2019 TRANSACTIONS OF THE VSB-TECHNICAL UNIVERSITY OF OSTRAVA CIVIL ENGINEERING SERIES 79