Legion Bridge in Prague - Assessment of Stone Arches

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