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Strojarsto 51(1) 27-37 (2009) I. ATIPOvI et. al., Numerical Procuedure for Two Beam... 27Numerical Procuedure for Two Beam... 2727

CODEN STJSAO ISSN 0562-1887

ZX470/1367 UDK 658.511/.512:65.012.11:004.421

Original scientic paper

The procedure for numerical analysis of the elastic impact of two beamsis presented. The beams are modelled by using the FEM technique, andfor time integration of dynamic equilibrium equation the modied linearacceleration method is applied. Special attention is paid to contact problemand to calculation execution. A short reiew of the energy conseration

principle based method for impact force determination is also gien.The procedure is illustrated with the case of collision of communication

bridge between two offshore platforms and the protectie construction,called arrestor. The results are compared to those obtained by energy

conseration principle based method. The study of damping inuence onsystem dynamic response is also added.

Numeriki postupak za analizu elastinog sudara dviju greda

Izornoznansteni lanak

Prikazan je postupak za numeriku analizu elastinog sudara dijugreda. Grede su modelirane koritenjem metode konanih elemenata,a za remensku integraciju jednadbe dinamike ranotee koritena jemodicirana metoda linearnog ubrzanja. Posebna pozornost poseena jekontaktnom problemu i proedbi prorauna. Ukratko je opisana metoda zaodreianje sile udara temeljena na principu ouanja energije. Primjena

postupka ilustrirana je na primjeru sudara komunikacijskog mosta izmeudiju platformi i zatitne konstrukcije, zane arestor. Dobieni rezultatiusporeeni su s rezultatima odreenim energetskim pristupom. Analiziran

je i utjecaj priguenja na dinamiki odzi sustaa.

Ivan ATIPOVI1), Nikola VLADIMIR1)andMarin RELJI2)

1) Fakultet strojarsta i brodogradnje Seuilitau Zagrebu (Faculty of Mechanical Engineeringand Naal Architecture, Uniersity of Zagreb)Iana Luia 5, HR - 10000 ZagrebRepublic of Croatia

2) Brodarski Institut d.o.o.

A. v. Holjeca 20, HR - 10020 Zagreb Republic of Croatia

Keywords

Elastic impact

Energy method

FEM

Linear acceleration method

Kljune rijei

Elastini sudar

Energijska metoda

MKEMetoda linearnog ubrzanja

Received (primljeno): 2008-02-20Accepted (prihvaeno): 2008-12-19

Numerical Procedure for Two Beam Elastic Impact Analysis

[email protected]

1. Introduction

The procedure for numerical analysis of the elasticimpact of two beams, based on linear accelerationmethod, is presented. Motiation for this work is foundin the practical engineering problem of analysing impact

response of the communication bridge between twooffshore platforms and protectie construction calledarrestor. It is common to nd two offshore facilities atone oil and gas exploitation place. The rst of them,which is usually xed, is used to maintain all technologyoperations needed for oil and gas exploitation. The secondone is used for crew accommodation. Efcient connectionof these facilities is achieed through a communicationbridge for crew and material transport, Figure 1. Thebridge interconnects xed and oating facilities. Thus,one support is xed and another one is moable. Whenfacilities mentioned are distant there is the possibility of

the bridge pitching into the sea. So, the task of the arrestoris to restrain the bridge without plastic deformation. There

are many types of arrestors and also a few calculationprocedures aailable, for example procedures based onenergy conseration principle [1], and some procedureswhich assume the plastic deformation of the arrestor. Theauthors prefer elastic calculation methodology, althoughit is rarely used, because when the arrestor is designed in

that way, it can be used in similar situations.

2. Numerical procedure

Two elastic beams, i.e. upper and lower beam, areunder consideration, Figure 2. One end of the upperbeam is sliding supported, and the other one is hingejointed. If sliding support is moed off, the upper beamis pitching and strikes the free end of the second beam.Another end of the lower beam is xed. This is the startof dynamic interaction of beams, i.e. alteration of contact

and non-contact stages. The basic equation of dynamicequilibrium yields [2]:

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28 I. ATIPOvI et. al., Numerical Procuedure for Two Beam... Strojarsto 51(1) 27-37 (2009)

Symbols/Oznake

Ac - cross-section area of contact element

-porina presjeka kontaktnog elementa

d - horizontal distance between lower beam supportand end- horizontalna udaljenost izmeu oslonca i krajadonje grede

E -Youngs modulus- Youngo modul elastinosti

Edef, a

- arrestor strain energy- energija deformacije arestora

Edef, b

-bridge strain energy- energija deformacije mosta

Fc - contact element load

- optereenje kontaktnog elementaF

din - dynamic contact force

- dinamika kontaktna sila

g - acceleration of graity- graitacijsko ubrzanje

h - ertical distance between the upper beam centreof graity at initial and instant at impact time,

- ertikalna udaljenost izmeu teita gornjegrede u poetnom trenutku i trenutku udara

hdin

- ertical distance between the upper beamcentre of graity at instant impact time and atinstant maximum deection instant

- ertikalna udaljenost izmeu teita gornjegrede u trenutku udara i maksimalnog progiba

i - time step, s- remenski korak

Ju - upper beam mass inertia moment

- moment tromosti mase gornje grede

Kf - exion coefcient

- koecijent eksije

lc - contact element length

- duljina kontaktnog elementa

ll - lower beam length

- duljina donje grede

lu - upper beam length

- duljina gornje grede

Mb -bridge bending moment

- moment saijanja mosta

MG, u

- upper beam weight moment- moment teine gornje grede

mi - modal mass for certain ibration mode

- modalna masa za odreeni oblik ibriranja

ml - lower beam mass

- masa donje grede

mu - upper beam mass- masa gornje grede

ql - lower beam distributed load

- distribuirano optereenje donje grede

qu - upper beam distributed load

- distribuirano optereenje gornje grede

t - time ariable- rijeme

t0 - time when upper beam is in

horizontal position- trenutak kada je gornja greda u

horizontalnom poloaju

wdin

- dynamic deection- dinamiki progib

yk, l

- degree of freedom of lower beam wherethere is contact

- stupanj slobode donje grede na mjestukontakta

yk, u

- degree of freedom of upper beam wherecontact is realized

- stupanj slobode gornje grede na mjestukontakta

[C] - damping matrix- matrica priguenja

[Cl] - lower beam global damping matrix

- globalna matrica priguenja donje grede

[Cu] - upper beam global damping matrix

- globalna matrica priguenja gornje grede

[K] - stiffness matrix- matrica krutosti

[Kc] - contact element stiffness matrix

- matrica krutosti kontaktnog elementa

[Kl

] - lower beam global stiffness matrix- globalna matrica krutosti donje grede

[Ku] - upper beam global stiffness matrix

- globalna matrica krutosti gornje grede

[M] - mass matrix- matrica mase

[Ml] - lower beam global mass matrix

- globalna matrica masa donje grede

[Mu] - upper beam global mass matrix- globalna matrica masa gornje grede

[P], [Q], [R] - auxiliary matrices- pomone matrice

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..{} - acceleration ector

- ektor ubrzanja

{l} - lower beam acceleration ector

- ektor ubrzanja donje grede

{u} - upper beam acceleration ector- ektor ubrzanja gornje grede

u - upper beam inclining angle

- kut nagiba gornje grede

u0

- upper beam initial inclining angle- poetni nagib gornje grede

i - non-dimensional damping coefcient for certain

ibration mode- bezdimenzijski koecijent priguenja zaodreeni oblik ibriranja

t - interal duration

- remenski interal

u

- upper beam angular acceleration- kutno ubrzanje gornje grede

i

- natural frequency of certain ibration mode- prirodna frekencija odreenog oblikaibriranja

u

- upper beam angular elocity- kutna brzina gornje grede

[S] - auxiliary matrices- pomone matrice

[] - natural ibration modes matrix- matrica prirodnih oblika ibriranja

{F (t)} - force ector- ektor sile

{Fl} - lower beam nodal forces ector

- ektor ornih sila donje grede

{Fu} - upper beam nodal forces ector

- ektor ornih sila gornje grede

{f}i+1 - auxiliary ector

- pomoni ektor

{} - displacement ector- ektor pomaka

{l} - lower beam displacement ector- ektor pomaka donje grede

{u} - upper beam displacement ector

- ektor pomaka gornje grede

{} - elocity ector- ektor brzine

{l} - lower beam elocity ector

- ektor brzine donje grede

{u} - upper beam elocity ector

- ektor brzine gornje grede

.

.

.

..

..

Figure 1.Schematic iew of bridge and arrestor

Slika 1.Shematski prikaz mosta i arestora

[K]{} + [C]{} + [M]{} = {F (t)} (1)

Global stiffness and mass matrices for upper andlower beam are dened according to [2], and global

damping matrices for upper and lower beam are denedaccording to [3]. Two types of nite elements are used;beam 4DOF nite element and bar 2DOF nite element.Beam elements are used for the upper and lower beamand the bar element is used for impact location modeling(contact element). The role of the contact element is thedescription of local behaior of both beams near theimpact place. Since the procedure intends to gie a globalresponse, the inuence of considered contact element onglobal deformations of the rest of the construction shouldbe minimized. It is achieed by a relatiely large stiffnessof contact element. This way, only the elastic deformationof contact element is allowed.

Figure 2.Disposition of upper and lower beam

Slika 2.Dispozicija gornje i donje grede

. ..

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During the pitching of the upper one, the consideredbeams are two independent bodies. Only during thecontact can these beams be treated as one structure. Itis also obious that the beams are not in contact frominstant impact time till instant resting time. Moreoer,

the interaction of considered beams consists of a numberof contact and non-contact periods. In other words,we can say that the contact of beams can sustain onlycompressie but not a tensile force. This fact should betaken into account when dening the system of motionequations. Therefore, by means of Eq. (1) two systemswith two possible stages of beams interaction shouldbe dened, i.e. the rst one without the contact andthe second one with contact. The rst stage, where twoindependent bodies are inherent, is dened by:

.

(2)

During contact, both beams behae as a uniquestructure, so the coupling should be introduced into asystem of equation (2). The coupling is achieed throughnew stiffness matrix, because inertial and damping forcestake effect on both beams independently. The new globalstiffness matrix is made by imposing the bar 2DOF

nite element stiffness matrix into the old one. So, thefollowing matrix [2]:

,(3)

is added up together with matching terms in the oldstiffness matrix. Now, the global stiffness matrix can beshown in the following form:

.

(5)

It is more suitable to replace Eq. (5) with the followingone:

. (6)

Thus, if the condition in Eq. (6) is satised, there iscontact between the beams. Otherwise the beams are in anon-contact stage.

2.1. Linear acceleration method

Numerical integration of Eq. (1) is done by using thelinear acceleration method [3]. This method assumeslinear acceleration in specied time interal. Accelerationector is determined according to:

,(7)

where:

.(8)

Integration matrix is dened by the followingformula:

.

(9)

Auxiliary matrices are dened in the following form:

(10)

velocity and displacement ector at time instant ti+1

.

(11)

The linear acceleration method shown here slightlydiffers from that gien in [3]. Although there thedisplacement is calculated rst, here it is necessaryto rst calculate the acceleration and then the elocityand displacement. Otherwise, numerical instability canoccur when ery low interal duration tis used, becauseof subtraction of ery close numbers. When the linearacceleration method is used in recommended form, it isnumerically stable.

2.2. Load

It is assumed that the external load is caused onlyby graitational force. Taking into account that cross-

(4)

Since contact and non-contact stages alternate, thecondition that indicates the starting and the ending of eachstage is the appearance and disappearance of compressie

load in contact element. So the condition yields:

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section of beams is assumed to be uniform, the weightis uniformly distributed per length. Distributed load ofupper beam:

(12)and for distributed load of the lower beam:

. (13)

2.3. Initial conditions

It is assumed that in instant initial time the lower beamis in a horizontal and the upper beam is in an inclinedposition. The initial inclination angle is

u0. This topology

is used because it is easy to satisfy the compatibilityof beam deections, but some other topology is alsoallowed. During pitching, the upper beam rotates aroundthe joint as a rigid body. The equation of motion has thefollowing form:

.(14)

The beam mass moment of inertia is calculated by:

(15)

and the upper beam weight moment by:

.

(16)

By substituting (15) and (16) into (14) and afterediting, the following expression is deried:

.

(17)

By integrating Eq. (17) the upper beam angularelocity is deried:

. (18)

Integration of Eq. (18) gies the alue of the upperbeam inclining angle during pitching:

. (19)

The initial inclination angle is calculated by using thefollowing formula:

. (20)

Then, the time when the upper beam is in horizontalposition:

(21)

and taking Eq. (18) into account, the angular elocity in

horizontal position:

. (22)

Initial conditions for the upper beam {u}

0and {

u}

0

are dened by using Eqs. (17), (20) and (22) and followthe next model:

(23)

where kiis the horizontal distance from a certain node toan upper beam joint.

For satisfying the motion equation in the initial timeinstant, deection is calculated by using the followingformula:

.(24)

The initial elocity of the lower beam is zero so onecan write:

. (25)

Initial deection:

, (26)

where {Fl}

0is the initial force ector of the lower beam.

3. Energy conservation principle approach

The impact force between to gien beams can becalculated by applying the energy conseration principle.In the instant initial time instant the upper beam has agraitational potential energy which is transformed intokinetic energy during the fall. After the impact, kineticenergy of the upper beam transforms into kinetic energyof the lower beam and the strain energy of the upperand the lower beam. The energy dissipation duringimpact, like friction or damping dissipation or plasticdeformations near the impact location, are not taken intoaccount. There are also two restrictions; it is assumedthat the stiffness of the upper beam is seeral timeshigher than the lower beam stiffness, so the upper beamstrain energy is negligible when compared to lower beamstrain energy. We also assume that there is zero elocityat maximum deection of the lower beam, so the kineticenergy of both beams equals zero. Taking into account

. ..

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the preiously mentioned simplication, we can presumethat graitation potential energy of the upper beamtransforms into the strain energy of the lower beam [1]

.

(27)

Bearing in mind Figure 2 one can write:

. (28)

Maximum of dynamic contact force is achieed whenmaximum w

din

. (29)

By introducing Eqs. (28) and (29) into (27) one canwrite:

, (30)

. (31)

This quadratic equation gies the dynamic deectionof the lower beam end

. (32)

Only the positie solution ts the physical backgroundso one can write

. (33)

4. Illustrative example

The collision of the communication bridge betweentwo offshore platforms and the arrestor is analysed. It isassumed that in the initial instant the bridge is leaned onboth ends, Figure 1. When the platforms are distant, thebridge starts to pitch and strikes the arrestor. The bridgeis a braced construction, and is made of high-tensile steel,Figure 3. Properties of the bridge are the following [4]:

Mass mb

= 12 t

Length lb

= 29 m

Second moment ofcross-section oer

Ib

= 0,01831 m4

Section modulus Wb

= 0,009978 m3

Youngs modulus Eb

= 2,1105MPa

Maximum bending stress al,b = 533 MPa

Figure 3.Bridge structure

Slika 3.Struktura mosta

The arrestor is a protectie construction consisting oftwo longitudinal beams and one transerse beam, Figure4. Geometrical properties of the arrestor I prole are thefollowing:

Height 300 mm

Breadth 1000 mmWeb breadth 20 mm

Flange thickness 24 mm

Second moment of cross-section oer 0,001038 m4

Section modulus 0,006918 m3

Cross-section area 0,05374 m2

Shear area 0,006 m2

Material properties of the arrestor:

Youngs modulus Ea = 2,1x105 N/mm2

Maximum bending stress al, a

= 533 N/mm2

Maximum shear stress al, a

= 355 N/mm2

Figure 4.Arrestor structure

Slika 4.Arestor

It is obious that both the bridge and the arrestor canbe treated as uniform beams. Properties of the arrestorused in numerical calculation and treated as uniformbeam are:

Mass ma= 11,7 t

Length la = 12 mBreadth b

a= 3,7 m

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Cross-section moment of inertia Ia = 0,002075 m4

Section modulus Wa = 0,01384 m3

Shear area As,a

= 0,012 mm2

Numerical analysis is done with 22 nite elements.

The FEM topology is shown in Figure 5. Simulationduration is 3 s and 90 000 time steps are necessary.Time step duration in non-contact stage is 510-5s andfor contact stage 2,510-5, respectiely. Contact elementlength is chosen to be 0,3 m, and its cross-section areaA

c= 0,04 m2. Assessment of damping matrix for bridge

and arrestor is based on critical damping alue withgien proportionality alue. Global damping matrix iscalculated by the following formula [3]:

. (34)

The non-dimensional damping coefcient alue is

taken to be i= 0,03.Arrestor end is on the position of node 8, and contact

position on the bridge is presented by node 16. Deectionof these nodes is shown in Figure 6. Critical cross-sectionof the arrestor is at the clamp position, i.e. node 1, and thecritical cross-section of the bridge is in the contact node,Figure 7. Shear stress of the arrestor at clamp positionis presented in Figure 8 and the contact element stressis shown in Figure 9. Comparison of deried numericalresults to those obtained by energy conseration approachis shown in Table 1.

Figure 5.FEM topology of bridge and arrestor

Slika 5.Topologija konanih elemenata mosta i arestora

Figure 6.Deection of arrestor end (node 8) and bridge at

contact position (node 16)Slika 6.Progib kraja arestora (or 8) i mosta na mjestukontakta (or 16)

Figure 7.Bending stress at critical cross-sections of arrestor(node 1) and bridge (node 16)

Slika 7.Saojno naprezanje u kritinim presjecima arestora(or 1) i mosta (or 16)

The assumption that the strain energy of the bridgeis negligible when compared to the strain energy of thearrestor is also checked. The strain energy of the arrestoris calculated according to:

(35)

and equalsEdef,a

= 149,2 kJ.

Figure 8.Shear stress of arrestor (node 1)

Slika 8.Smino naprezanje u arestoru (or 1)

Figure 9.Contact element stress

Slika 9.Naprezanje u kontaktnom elementu

t,s

,m

t,s

,MPa

t,s

,MP

a

t,s

,MPa

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Table 1.Results comparison

Tablica 1.Usporedba rezultata

(1)

FEM / Metodakonanih elemenata

(2)

Energy approach /Energijski pristup

(1)/(2)

wdin,a

0,691 m 0,628 m 1,100

din,a

492 MPa 412 MPa 1,194

din,a

59 MPa 40 MPa 1,475

din,b

295 MPa 219 MPa 1,347

The strain energy of the bridge [1] is calculated by thefollowing formula:

(36)

and when arrestor deection is maximum, it equalsEdef,b= 4,4 kJ. It is obious that the bridge strain energy makesonly 3,0 % of the arrestor strain energy. It is also wellknown that in steel constructions ibrations analysisdamping forces are negligible when compared to inertialand restoring forces. These constructions ibrate in phase,and when deection is maximal the ibration elocity isrelatiely small.

The agreement of the results is acceptable in case ofdeection and stresses of the arrestor. Large discrepanciesare found in case of bending stress of the bridge. Bridgeshear stress is not calculated since shear stress forbraced construction is negligible. Calculation of naturalfrequencies and natural modes of uncoupled and coupledibrations has been done. Natural modes are shown inFigures 10 to 12. Natural frequencies and natural modesare presented in Tables 2 and 3.

Figure 10.Natural modes of arrestorSlika 10. Prirodni oblici ibriranja arestora

Figure 11.Natural modes of bridge

Slika 11.Prirodni oblici ibriranja mosta

It can be seen in Figure 6 that during the interaction ofthe bridge and the arrestor, from 0,77 s to 1,44 s, there is

no contact. During that time the arrestor is ibrating freely,and from 0,77 s to 1,24 s makes one oscillation whichtakes 0,47 s. The duration of the mentioned oscillation isnearly a natural period of the arrestor for the rst naturalmode, Figure 10. Therefore, one can conclude that thearrestor is oscillating in rst mode in non contact-stage.A similar conclusion can be deried from looking intoFigures 7 and 8. Figure 7 shows that stress leel arieswith period 0,12 s to 0,14 s, irrespectie of contact or non-contact stage. This period is close to the second naturalperiod of the bridge, Figure 11, and the second naturalmode of coupled ibrations of bridge and arrestor, Figure

12. One can conclude that in bridge ibrations the secondnatural mode is dominant.

Figure 12.Natural modes of bridge and arrestor coupledmotion

Slika 12. Prirodni oblici spregnutih ibracija mosta i arestora

5. Conclusion

The presented numerical procedure for the analysisof the elastic impact of two beams seems to be moresuitable for detailed structural analysis, since the energy

approach principle is more conenient for standardengineering practice. The energy approach is especially

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appropriate in design phase. Comparison of the resultsshowed good agreement between deections and stressesat critical arrestor cross-section, but large discrepanciesoccurred in stress calculation at critical cross-sectionof bridge. There are two reasons why this is the case.

The rst is that critical cross-section of the arrestor israther distant from contact location, where large localstresses and deformations are present. Consideringthe bridge, the situation is opposite, since the criticalcross-section is at contact location. It is well knownthat energy approach cannot capture local deformationsand stresses, and therefore it gies lower stress leel forthe arrestor and particularly for the bridge. The secondone is that energy approach does not distinguish naturalibration modes and their inuence on stress shape. Itshould be emphasized that higher ibration modes haea large inuence on stress leel for rigid structures, for

example the bridge. That is the explanation why energyapproach gies signicantly lower stress leel for rigidstructures compared with numerical procedure. Thesetwo approaches, i.e. the numerical approach and theenergy approach support each other.

Table 2.Natural frequencies

Tablica 2.Prirodne frekencije

Natural frequencies / Prirodne frekencije, rad/s

n arrestor / arestor bridge / most

bridgeand

arrestor

/ most iarestor

1 13,7 0,0 5,6

2 92,9 55,9 45,8

3 270,9 181,1 84,4

4 546,1 377,9 181,1

5 923,5 646,5 251,8

Table 3.Natural periods

Tablica 3.Prirodni periodi

Natural periods / Prirodni periodi, s

n arrestor / arestor bridge / mostbridge and

arrestor / most iarestor

1 0,458 / 1,130

2 0,068 0,112 0,137

3 0,023 0,035 0,074

4 0,012 0,017 0,035

5 0,007 0,010 0,025

Appendix a. Study of damping inuence on system

dynamic response

The inuence of damping on bridge and arrestordeformations and stresses is analysed. As mentionedbefore, damping matrices for bridge and arrestor arederied by taking critical damping for all ibration modesinto account. Non-dimensional damping coefcient forcalculation is taken to be

i= 0,03. Recommended alue

of this coefcient is between 0,02 and 0,05 for standardibration analyses. Haing in mind that the impact loadis present here, i.e. high elocity during the structuredeformation, it is necessary to conduct a more detailedanalysis of damping inuence on structural response.Therefore, the simplied model of bridge and arrestorhas been made, Figure A1. The calculation is executedseeral times with different alues of non-dimensionaldamping coefcient: 0; 0,01; 0,015; 0,02; 0,03 and0,04. Bridge and arrestor deections in correlation withdifferent alues of non-dimensional damping coefcientsare shown in Figures A2 to A7. One can conclude that thedamping coefcient does not hae signicant inuenceon bridge and arrestor deection. Damping coefcientinuence is signicant for bridge stresses, Figures A8 toA13. Time distribution of stress is similar for

i= 0,0 and

i= 0,1. Large discrepancies can be noticed at

i= 0,0015.

A further increase of the damping coefcient does notdecrease the stress leel, i.e. for

i= 0,02,

i= 0,03 and

i

= 0,04 the stress leel is approximately equal. The bridgestress leel is rather inuenced by local deformations, i.e.higher ibration modes. At i= 0,0 and i= 0,01 highermodes are still undamped and signicantly increasestress leels. Finally, one can conclude that the dampingdenition described in this chapter has a large inuenceon higher ibration modes, but its inuence on lowermodes is negligible.

Figure A1.Simplied topology of bridge and arrestor

Figure A1.Pojednostaljena topologija mosta i arestora

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Figure A2.Deection of arrestor end and bridge contact point,

i= 0

Slika A2.Progib kraja arestora i mjesta kontakta na mostu,

i= 0


i= 0,01


i= 0,01


i= 0,015


i= 0,015


i= 0,02


i= 0,02


i= 0,03


i= 0,03


i= 0,04


i= 0,04

Figure A8.Bending stress at critical cross-section of bridgeand arrestor,

i= 0,0

Slika A8.Saojno naprezanje u kritinim presjecima mosta iarestora,

i= 0,0


i= 0,01


i= 0,01

arrestor (node 3) / arestor (or 3) bridge (node 6) / most (or 6)

t,s

,m


t,s

,m


t,s

,

m


t,s

,m


t,s

,m


t,s

,m


t,s

,

MPa


t,s

,

MPa

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i= 0,015


i= 0,015


i= 0,02


i= 0,02


i= 0,03


i= 0,03


i= 0,04


i= 0,04

REFERENCES

[1] ALFIREvI, I.: Strength of materials II, Technicalmechanics library, Book-5, Golden marketing, Zagreb,2005.

[2] SENJANOvI, I.:Finite element method in the analysisof ship structures, Uniersity of Zagreb, Zagreb, 1998.

[3] SENJANOvI, I.: Ship vibrations, Part 3, Textbook,Uniersity of Zagreb, Zagreb, 1981.

[4] Tyne Gangway (Structures) Ltd., Howdon Lane, Wallsend-on-Tyne, England.


t,s

,

MPa


t,s

,M

Pa


t,s

,

MPa


t,s

,

MPa

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