Building Materials and Structures

8/13/2019 Building Materials and Structures

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YU ISSN 05 43-0 798 UDK: 06.055.2:62-03+620.1+624.001.5(497.1)=861

2013.GODINA

LVI

GRAEVINSKIMATERIJALI IKONSTRUKCIJE

BUILDINGMATERIALS AND

STRUCTURESA SO PI S Z A I S TR A I V AN J A U O BL AS TI MA T ER I J A L A I KO N ST R U K C I J A

J O U R N A L F O R R E S E A R C H OF M A T E R I A L S A N D S T R U C T U R E S

DRUTVO ZA ISPITIVANJE I ISTRAIVANJE MATERIJALA I KONSTRUKCIJA SRBIJE

SOCIETY FOR MATERIALS AND STRUCTURES TESTING OF SERBIA

DDIIMMKK

3


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Odlukom Skuptine Drutva za isp iti van je materij ala ikonstrukc i ja, odrane 19. aprila 2011. godine u Beogradu,promenjeno je ime #asopisa Materijali i konstrukcijei od sada%e se #asopis publikovati pod imenom Gra"evinski materijali ikonstrukcije.

According to the decision of the Assembly of the Society fo rTestin g Materials and Struc tures, at the meeting held on 19

April 2011 in Belgrade the name of the Journal Materijali ikonstrukcije (Materials and Structures) is changed intoBuilding Materials and Structures.

Professor Radomir FolicEditor-in-Chief


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DRUTVO Z'ISPITIV'NJE I ISTR'IV'NJE M'TERIJ'L' I KONSTRUKCIJ'SRBIJES O C I E T Y F O R M' T E R I' L S ' N D S T R U C T U R E S T E S T I N G O F S E R B I'

GGRRAAEEVVIINNSSKKII BBUUIILLDDIINNGGMMAATTEERRIIJJAALLIIII MM##TTEERRII##LLSSAANNDD

KKOONNSSTTRRUUKKCCIIJJEE SSTTRRUUCCTTUURREESS'S O P I S Z A I S T R' I V'N JA U O BL'S T I M'T E R I J'L' I K O N S T R U K C I J' JOURN'L FOR RESE'RCH IN THE F IELD OF M'TERI'LS 'ND STRUCTURES

INTERNATIONAL EDITORIAL BOARD

ProfessorRadomir Foli%, Editor in-ChiefFaculty of Technical Sciences, University of Novi Sad, Serbia

Fakultet tehni#kih nauka, Univerzitet u Novom Sadu, Srbijae-mail:[email protected]

ProfessorMirjana Maleev, Deputy editorFaculty of Technical Sciences, University of Novi Sad,SerbiaFakultet tehni#kih nauka, Univerzitet u Novom Sadu, Srbijae-mail: [email protected]

DrKsenija Jankovi%Institute for Testing Materials, Belgrade, SerbiaInstitut za ispitivanje materijala, Beograd, Srbija

DrJose Adam, ICITECHDepartment of Construction Engineering, Valencia,Spain.

Professor Radu Banchila

Dep. of Civil Eng. Politehnica University ofTemisoara, Romania

Professor Dubravka Bjegovi%Civil Engineering Institute of Croatia, Zagreb, Croatia

Assoc. professor Meri CvetkovskaFaculty of Civil Eng. University "St Kiril and Metodij,Skopje, Macedonia

Professor Michael FordeUniversity of Edinburgh, Dep. of Environmental Eng.UK

Dr Vladimir GocevskiHydro-Quebec, Motreal, Canda

Professor Miklos IvanyiUniversity of Pecs, Faculty of Engineering,Hungary.

Professor Asterios LioliosDemocritus University of Thrace, Faculty of CivilEng., Greece

Predrag Popovi%Wiss, Janney, Elstner Associates, Northbrook,Illinois, USA.

Professor Tom SchanzRuhr University of Bochum, Germany

Professor Valeriu StoinDep. of Civil Eng. Poloitehnica University ofTemisoara, Romania

Acad. Professor Miha Tomaevi%, SNB and CEI,Slovenian Academy of Sciences and Arts,

Professor Mihailo Trifunac,Civil Eng.Department University of Southern California, LosAngeles, USA

Lektori za srpski jezik: Dr Milo Zubac, profesorAleksandra Borojev, profesor

Proofreader: Prof. Jelisaveta afranj, Ph DTechnic)l editor: Stoja Todorovic, e-mail: [email protected]

PUBLISHER

Society for Materials and Structures Testing of Serbia, 11000 Belgrade, Kneza Milosa 9Telephone: 381 11/3242-589; e-mail:[email protected], veb sajt: www.dimk.rs

REVIEWERS: All papers were reviewedCOVER: Izgradnja garae "Pionirski park" u Beogradu (foto prof. dr Milan Maksimovi%)

Construction of garage "Pionirski park" in Belgrade (photo prof. dr Milan Maksimovic)

Financial supports: Ministry of Scientific and Technological Development of the Republic of Serbia
mailto:e-mail:[email protected]:[email protected]:[email protected]:e-mail:[email protected]://www.dimk.rs/http://www.dimk.rs/mailto:e-mail:[email protected]:[email protected]:[email protected]:e-mail:[email protected]


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YU ISSN 0543-0798 GODINA LVI - 2013.

DRUTVO Z'ISPITIV'NJE I ISTR'IV'NJE M'TERIJ'L' I KONSTRUKCIJ'SRBIJES O C I E T Y F O R M' T E R I' L S ' N D S T R U C T U R E S T E S T I N G O F S E R B I'

GGRRAAEEVVIINNSSKKII BBUUIILLDDIINNGG

MMAATTEERRIIJJAALLIIII MM##TTEERRII##LLSSAANNDDKKOONNSSTTRRUUKKCCIIJJEE SSTTRRUUCCTTUURREESS'S O P I S Z A I S T R' I V'N JA U O BL'S T I M'T E R I J'L' I K O N S T R U K C I J' JOURN'L FOR RESE'RCH IN THE F IELD OF M'TERI'LS 'ND STRUCTURES

S#DR#J

Dragan D. MILAINOVI(Mila SVILARNataa MRALaura TUZA

Branislav NOVKOVLjubomir MILAINOVI(VISKOELASTINA ANALIZA PRETHODNONAPREGNUTIH BETONSKIH GREDNIH NOSAAMETODOM KONANIH TRAKAOriginalni nau*ni rad ...............................................

Duan GRDI(Nenad RISTI(Gordana TOPLI,I((UR,I(UTICAJ DODATKA RECIKLIRANE GUME IRECIKLIRANOG STAKLA NA PROMENU BRZINEULTRAZVUKA U BETONUOriginalni nau*ni rad .................................................

Aleksandar PROKI(

Martina VOJNI(PUR,ARLAMINIRANI TANKOZIDNI NOSAI DRUGIDEOOriginalni nau*ni rad ...............................................

Uputstvo autorima ...................................................

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CONTENTS

Dragan D. MILASINOVICMila SVILARNataa MRDJALaura TUZA

Branislav NOVKOVLjubomir MILASINOVICVISCOELASTIC ANALYSIS OF PRESTRESSEDCONCRETE GIRDERS BY THE FINITE STRIPMETHODOriginal scientific paper ..........................................

Dusan GRDICNenad RISTICGordana TOPLICIC CURCICEFFECTS OF ADDITION OF RECYCLED RUBBER

AND RECYCLED GLASS ON THE VARIATIONS OFULTRASONIC VELOCITY IN CONCRETEOriginal scientific paper............................................

Aleksandar PROKIC

Martina VOJNIC PURCARLAMINATED THIN-WALLED BEAMS - SECONDPARTOriginal scientific paper ...........................................

Preview report ..........................................................

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GRAEVINSKI MATERIJALI I KONSTRUKCIJE 56 (2013) 3 (3-28)BUILDING MATERIALS AND STRUCTURES 56 (2013) 3 (3-28)

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VISKOELASTINA ANALIZA PRETHODNO NAPREGNUTIH BETONSKIH GREDNIHNOSAA METODOM KONANIH TRAKA

VISCOELASTIC ANALYSIS OF PRESTRESSED CONCRETE GIRDERS BY THE

FINITE STRIP METHOD

Dragan D. MILAINOVI

Mila SVILARNataa MR#ALaura TUZABranislav NOVKOVLjubomir MILAINOVI

ORIGINALNI NAU!NI RADORIGINAL SCIENTIFIC PAPER

UDK: 624.012.36.072.2 = 861

1 UVOD

U klasi$noj teoriji elasti$nosti pretpostavlja se da suveze izme&u napona i deformacija linearne i da sunezavisne od vremena. Superpozicija je u vaenju i zanapone i za deformacije. Zavisnost izme&u napona ideformacija jeste nezavisna - od trenutka kada je optere-

'enje uvedeno i od trajanja dejstva optere'enja [1].Poznato je da ponaanje materijala vie ili manje odstupa od pretpostavki uvedenih u teoriju elasti$nosti.Njihovo ponaanje zavisi od toga kako se s vremenommenjaju naponi i deformacije. Eksperimenti s betonom,kao i s nizom drugih konstrukcionih materijala, ukazali suna to da se materijali druga$ije ponaaju u slu$aju naglonaneenog optere'enja, a druga$ije u slu$aju dejstva

Prof. dr Dragan D. Milainovi', Univerzitet u Novom Sadu,Gra&evinski fakultet Subotica, Kozara$ka 2a,e-mail: [email protected] Svilar, student doktorskih studija, Univerzitet u Novom

Sadu, Gra&evinski fakultet Subotica, Kozara$ka 2a,e-mail: [email protected] Mr&a, student doktorskih studija, Univerzitet uNovom Sadu, Gra&evinski fakultet Subotica, Kozara$ka 2a,e-mail: [email protected] Tuza, student doktorskih studija, Univerzitet u NovomSadu, Gra&evinski fakultet Subotica, Kozara$ka 2a,e-mail: [email protected] Novkov, student doktorskih studija, Univerzitet uNovom Sadu, Fakultet tehni$kih nauka, Novi Sad, TrgDositeja Obradovi'a 6, e-mail: [email protected] Milainovi', student doktorskih studija, Univerzitetu Novom Sadu, Fakultet tehni$kih nauka, Novi Sad, TrgDositeja Obradovi'a 6, e-mail: [email protected]

1 INTRODUCTION

In the classical theory of elasticity it is supposed thatthe stress-strain relationships are linear and independenton time. Superposition is valid for both stresses andstrains. The relationships between stresses and strainsare independent from the moment when the load is

introduced and during the effect of loading [1].We know that materials behaviour deviates more or

less from the hypotheses offered in the elasticity theory.Their behaviour depends on the change of stresses andstrains with time. Experiments made with concrete and anumber of other construction materials point out theybehave differently in the case when the stress is appliedsuddenly, and when the permanent load is of long-term

Dragan D. Milainovi', PhD, University of Novi Sad, Facultyof Civil Engineering in Subotica, Kozara$ka 2a,e-mail: [email protected] Svilar, PhD student, University of Novi Sad, Faculty of

Civil Engineering in Subotica, Kozara$ka 2a, e-mail:[email protected] Mr&a, , PhD student, University of Novi Sad, Facultyof Civil Engineering in Subotica, Kozara$ka 2a,e-mail:[email protected] Tuza, PhD student, University of Novi Sad, Faculty ofCivil Engineering in Subotica, Kozara$ka 2a,e-mail:[email protected] Novkov, PhD student, University of Novi Sad,Faculty of Technical Sciences, Novi Sad, Serbia, TrgDositeja Obradovi'a 6,e-mail: [email protected] Milainovi', PhD student, University of Novi Sad,Faculty of Technical Sciences, Novi Sad, Trg DositejaObradovi'a 6, e-mail: [email protected]
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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4

trajnog tereta neprekidnog karaktera. Za ostvarenjedeformacije koja !e se ostvariti za kra!e vreme, dakle bre, potreban je ve!i napon, to vai i obrnuto. Vezaizme$u napona i deformacije zavisi od brzine napona ibrzine deformacije. U toku eksperimenata moe sezapaziti da se napon menja u slu%aju nepromenjenedeformacije i da se isto deava sa deformacijom uslu%aju nepromenjenog napona.

Teorija linearno viskoelasti%nih materijala jeste deoreologije koja se bavi svim problemima u pogleduelasti%nosti i viskoznosti konstrukcijskih materijala.Reologija, kao nauka, daje nam mogu!nost zasastavljanje i reavanje diferencijalnih jedna%inaviskoelasti%nih tela [2]. Tela koja imaju elasti%ne iviskozne osobine zovu se viskoelasti%na tela. Zaopisivanje njihovog reolokog ponaanja mi ih moemosmatrati kombinacijom dva tela: idealno elasti%nog tela iidealne te%nosti. Elasti%ne osobine obrazuju se poHukovom zakonu, dok te%nost prati Njutnov zakon.Paralelna kombinacija ovakvog linearno elasti%nog tela iidealne te%nosti naziva se viskoelasti%an materijal iliKelvinovo telo (dokazano u nizu eksperimenata). ZaKelvinovo telo karakteristi%no je da promene deformacija

tee grani%noj vrednosti pri konstantnim naponima (zarazliku od te%nosti, gde su deformacije neograni%ene).Obim reolokih studija danas se brzo iri na problemecikli%ke varijacije napona [3-5].

Za reavanje stati%kih viskoelasti%nih problemaprednapregnutih betonskih grednih nosa%a, u ovom raduuveden je metod kona%nih traka. Razlog za uvo$enje togmetoda jeste to to je on u reavanju nekoliko tipovaprakti%nih problema znatno bri pri reavanju od mnogoobuhvatnijeg i prilagodljivijeg metoda kona%nihelemenata.

U radu su date ire teorijske osnove za ravno iprostorno stanje napona potrebni za problematikuanaliziranu metodom kona%nih traka u datim primerima.

Primeri predstavljaju analizu jednostavnijeg, jednoaksi-alnog stanja napona, koji se metodom kona%nih trakamogu analizirati bez ograni%enja irih teorijskih osnova.

2 VISKOELASTINE JEDNAINE BETONAMETODOM KONANIH TRAKA

2.1 Linearno te#enje betona

Beton se svrstava u grupu materijalakoji ima najiruprimenu u konstrukterskoj praksi gra$evinarstva. Onposeduje sloene reoloke osobine koje se opisujusloenim konstitutivnim jedna%inama, u zavisnosti odtipa i namene konstrukcije, temperature i vlanosti

sredine, njegove starosti pri optere!enju, duinevremenskog intervala u kojem deluje optere!enje itd.Pored trenutnih (elasti%nih, elastoplasti%nih i plasti%nih)deformacija koje nastaju odmah posle promene stanjanapona, u ovom materijalu se pojavljuju, pri dugotrajnimnaponskim stanjima, tokom vremena i deformacije usledte%enja.

Ovde se daje jedna formulacija grani%nih zadatakateorije betonskih konstrukcija imaju!i u vidu slede!epretpostavke:

beton se posmatra kao homogen ili homogen poslojevima i izotropan materijal;

trenutne deformacije su elasti%ne;

veze izme$u napona i deformacija su linearne i

character. For the realization of strain which is to happenin a short period of time, that is to say more quickly, ahigher stress is necessary, and vice versa. Therelationship between stress and strain depends on boththe rate of stress, and rate of strain. In the course of theexperiments, it could be noticed that the stress altersduring unchanged strain and that the same thinghappens to the strain during unchanged stress.

The theory of the linearly viscoelastic materials is a

part of rheology, which deals with all the problemsconnected to elasticity and viscosity of constructionmaterials. Rheology, as a science, gives us an opportu-nity to assemble and process differential equations ofthe viscoelastic bodies [2]. The bodies which haveelastic and viscous feature are called viscoelasticbodies. To describe their rheological behaviour, we canconsider these bodies as combinations of these twoelements: perfectly elastic body and ideal liquid. Theelastic feature is formed by the Hooke's law, while theliquid is the subject of the Newton's law. The parallelcombination of such linear elastic body and ideal liquid iscalled viscoelastic solid material or Kelvin's body, whichhas been proved in a number of experiments. It ischaracteristic for the Kelvin's body that the time-depen-dent changes of strains under the constant stresses tendto reach a certain limit values (as opposed to the liquid,where the strains are unlimited). The scope of rheo-logical studies are expands today rapidly to the problemsof cyclic variation of stresses [3-5].

In this paper the finite strip method has introduced inthe study of quasy-static viscoelastic behaviour ofprestressed concrete girders. The reason for the intro-duction of this method lies in the fact that resolving ofseveral classes of practical problems it is much faster incomparison with the more comprehensive and adaptablefinite element method.

This paper provides the theoretical basis required forthe problems analysed by finite strip method in theexamples given. Exemples presents analysis of simplerstress state, whitch can be analysed with finite stripmethod without limitation of wider theoretical basis.

2 FINITE STRIP VISCOELASTIC EQUATIONS OFCONCRETE

2.1 Linear creep of concrete

Concrete belongs to the group of materials whichhave the most widespread application in the constructionpractice of civil engineering. It has complex rheologicalcharacteristics which are described by complex con-stitutive equations, depending upon the type and the

assignation of the structure, the temperature and therelative humidity of the environment, its age whileloaded, the length of time intervals in which the loadinghas its effect etc. Along with the instantaneous strains(elastic, elastoplastic and plastic) that occur immediatelyafter the alteration of the state of stress, we find strainscaused by creep in concrete as well, in the case ofsustained load.

We are giving a formulation of limit states in theory ofconcrete structures, bearing in mind the followingpresuppositions:

concrete is viewed as homogeneous as a whole orin layers, and as isotropic material,

the instantaneous strain is elastic,


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5

formuliu se kao algebarske; reoloke osobine i parametri se tokom vremena

menjaju.Ove pretpostavke imaju opravdanja za analizu stanja

napona i deformacija velikog broja konstrukcija zaspoljanje uticaje koji se javljaju u fazi gra,enja ieksploatacije [6].

2.2 Konstitutivne viskoelasti*ne jedna*ine za beton

Ponaanje materijala koje se menja u toku vremenamoe se ispitivati i eksperimentalno. Pod dejstvomstalnog optere%enja nanetog u odre,enom trenutku ,eksperimentalnim putem se odre,uju funkcije te#enja.Funkcija te#enja za beton, u slu#aju linearne zavisnostiizme,u deformacije i naprezanja, jeste

the relationship between stress and strain is linearand it is algebraically formulated,

rheological features and parameters are changedin time.

These pressupositions justify the analysis of thestate of stress and strain for a large number of structuresfor external influences that occur in the building andexploatation phase [6].

2.2 Constitutive viscoelastic equations of concrete

The behaviour of the material which changes withtime can be experimentally examined. We canexperimentally determine the creep function of concreteunder constant load put at a certain moment . Thecreep functions of concrete in the case of linearrelationship between the stress and strain is

( ) ( ) ( ), ,t t F t t = . (1)

Svaki proizvoljan zakon optere%enja $=$(t) moe seprikazati u vidu zbira beskona#no malih konstantnih

optere%enja %$i nanetih u trenucima i i definisati kao

Every arbitrary loading $=$(t) can be represented asthe sum of infinitesimal constant loads %$iapplied at the

moments i , and defined as

( ) ( )1

limn

i in

i

t g

=

= , (2)

gde je: where

( ) 0,

1,

i

i

i

tg

t

=

. (3)

Unose%i (2) u (1) i zamenjuju%i zbir integralom,dobijamo

By insertion of (2) into (1) and substitution withintegral, we obtain

( )

( )

( ) ( ) ( )0

0 0, ,

t

t

d

t F t d t F t t d

= + , (4)gde je

0t po#etak nanoenja optere%enja.

Stanje napona i deformacija analizira se numeri#kimreavanjem integralne jedna#ine (4), ili prevo,enjemintegralne veze na priblinu algebarsku. Uvedemo li zafunkciju te#enja izraz

where t0represents the beginning of loading.The state of stress and strain is analysed through

numerical solution of integral equation (4), or bysubstitution of the integral relationship with approximatealgebraic one. If we introduce the following expressionfor the function of creep

( ) ( )0

0

0 28

,1,

t tF t t

E E

= + , (5)

u kojem je prvi #lan na desnoj strani elasti#ni deo, a

drugi odloeni ili zakasneli, integralnu jedna#inu (4)moemo formulisati u ekvivalentnom algebarskom obliku

in which the first element on the right side is elastic part,

and the second is delayed part, then we can formulatethe integral equation(4) in equivalent algebraic form

( ) ( )

( ) ( )

( ) ( )0 0 0 0 00 28 0

1 , 1 , ,t tE

t t t t t t t E E E

= + + +

. (6)

Jedna#ina (6) poznata je kao algebarska jedna#ina,a ona je ujedno i opta algebarska, jer se iz nje mogudobiti sve ostale algebarske veze. U njoj su:

t starost betona u posmatranom vremenu;

0

t starost betona u momentu optere%enja;

( )0,t t koeficijent te#enja;

The equation (6) is known as algebraic equation.This equation is at the same time general one, since allthe other algebraic relationship can be derived from it. Itcomprises:

t the age of concrete in the observed periodof time,

t0 the age of concrete at the moment of


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( )t razlika napona u posmatranom vremenui momentu optere%enja;

( )0,t t koeficijent starenja;

0

E po#etni modul elasti#nosti;

28E modul elasti#nosti nakon 28 dana;

loading,&(t,t0) creep coefficient,'$(t) the difference of stress in the observed

period of time and at the moment ofloading,

((t,t0) ageing coefficient,E0 the initial modulus of elasticity,E28 the modulus of elasticity after 28 days,

( ) ( )

( )

( )( )

( )0

1

0 00

28 0

, 1, 1 ,

,

t

t

t t d Et t t d

t d E t t

= +

, (7)

( ) ( ) ( )0t t t = . (8)

Ako uvedemo parametre: If we introduce the following parameters:

( ) ( ) ( ) ( ) ( )01 0 2 0 028

1 , , 1 , ,E

K t t t K t t t t tE

= + = + , (9)

izraz (6) moemo napisati u obliku we can write the expression (6) as

( ) ( ) ( )

( ) ( )

1 20

0 0

K t K tt t t

E E = + . (10)

U optem slu#aju anizotropnog tela, broj elasti#nihkonstanti u vezama oblika (1) treba da bude 21. Uizotropnom telu broj konstanti smanjuje se na samo dve.Pri postavljanju zavisnosti izme,u napona i deformacija,koristi%emo se principom nezavisnosti dejstva sila iuoptiti jedna#inu (1) za slu#aj troosnog stanjanaprezanja.

In the general case of an anisotropic body, thenumber of elastic constants in relationships similar to (1)should be 21. With isotropic body the number ofconstants is reduced to only two. To establish therelationships between stress and strain, we should usethe principle of independence of force action andgeneralize the equation (1) for the case of the three-dimensional state of stress.

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( )

( ) ( )

( ) ( )

1 2

1 2

1 2

3

3

3

, , ,

, , ,

, , ,

, ,

, ,

, .

x x y z

y y x z

z z x y

xy xy

yz yz

xz xz

t t F t F t

t t F t F t

t t F t F t

t F t

t F t

t F t

= +

= +

= +

=

=

=

(11)

U izrazima (11) uzeta je u obzir izotropija tela koja seizraava ravnopravno%u pravaca , y i z . Moe se

zapaziti da je veli#ina( )1

1

,F t analogna modulu

elasti#nosti E u izotropnom elasti#nom telu. Odnos

( )( )

2

1

,

,

F t

F t

analogan je Poisson-ovom koeficijentu , a

veli#ina( )3

1

,F t

modulu klizanjaG .

Kao i u teoriji elasti#nosti, gde je vaio odnos

( )2 1

EG

=

+, moe se i ovde definisati analogna

zavisnost

In the equations (11) the isotropy of the body whichis expressed through equality of directions: ,y andis taken into account. It can be noted that the value

( )1

1

,F t is analogous to the modulus of elasticity E in

the isotropic elastic body. The ratio ( )( )

2

1

,

,

F t

F t

is

analogous to the Poissons coefficient , and the value

( )3

1

,F t

is analogous to the shear modulus G .

As in the theory of elasticity, where the expression

( )2 1

EG

=

+ was valid, we can define analogous

dependence here as well


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7

( )

( )

( )

( )

1

3 2

1

1

,1

, ,2 1

,

F t

F t F t

F t

=

+

, (12)

odakle je From this it follows that

( ) ( ) ( )3 1 2, 2 , ,F t F t F t = + . (13)

Na ovaj na#in, od tri funkcije ( )1 ,F t , ( )2 ,F t i

( )3 ,F t nezavisne su samo dve, a tre%a je preko njihjednozna#no odre,ena jedna#inom (13).

Konstitutivne jedna#ine (11) dalje %emo napisati zaslu#aj dvoosnog stanja naprezanja i koritenjemalgebarskih veza (10), a uz pretpostavku da su Poisson-ovi koeficijenti jednaki i nepromenjivi tokom vremena. Umnogobrojnim eksperimentima dobijeni su rezultati ovrednostima ovih koeficijenata, u zavisnosti od sastavabetona i uslova u kojima su istraivanja vrena.

Pomenuta pretpostavka ima opravdanja, posebno zaprakti#ne analize.

Two functions ( )1 ,F t , ( )2 ,F t are independent,

while the third one ( )3 ,F t is through them uniformlydetermined by the equation (13).

We shall write the constitutive equations (11) for thecase of plane stress by applying algebraic relationships(10) assuming that Poissons coefficients are equal andconstant. Through many experiments were obtained thevalues of these coefficients. They depend on thestructure of concrete and conditions in which theexperiments were carried out. The above mentioned

assumption is justified, especially for practical analyses.

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( )

0 0 1 0 2 0

0 0 1 0 2 0

0 1 0 2 0

/ / ,

/ / ,

2 1 / 2 1 / .

x x y x y

y y x y x

xy xy xy

t t t K t E t t K t E

t t t K t E t t K t E

t t K t E t K t E

= + = +

= + + +

(14)

Ako uvedemo vektore: If we introduce vectors:

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

0 0 0 0

,

,

,

T

x y xy

T

x y xy

T

x y xy

t t t t

t t t t

t t t t

= = =

!

!

(15)

tada jedna#ine (14) moemo napisati u matri#nom obliku then we can write the equation (14) in matrix form

( ) ( ) ( ) ( )0 0t t t t = + C ! C ! , (16)

gde su: where

( )

( )

( )10

0

1 0

1 0

0 0 2 1

K tt

E

= +

C , (17)

( )

( )

( )2

0

1 0

1 0

0 0 2 1

K tt

E

= +

C . (18)

Iz izraza (16) moemo izraziti i pove%anje napona On the basis of expression (16) we can express theincrease of stress

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )1 1 0 0 0 0t t t t t t t t t t = = ! C C C ! D D ! , (19)

gde su: where:


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8

( )

( ) ( )

( ) ( )

( )( )

2 2

2 2 0

2

1/ 1 / 1 0

/ 1 1/ 1 0

0 0 1/ 2 1

Et

K t

=

+

D, (20)

( )0

1

11

t

= D

. (21)

2.3 Jedna*ine ravnotee za linearno te*enje betonametodom kona*nih traka

Po,imo od opteg izraza za energiju deformacijehomogenog materijala

2.3 Finite strip equations of balance for linearcreep of concrete

Let us start with the general equation for the strainenergy of homogeneous material,

( ) ( ) ( ) ( )01 1

2 2

T T

m s

A A

U U U t t dA t t dA= + = + ! " M . (22)

Zadrimo li se samo na linearnim problemima, tada,koriste%i izraz (19), moemo definisati promene energijadeformacija:

If we stop solely at the linear problems, then we candefine the strain energy alterations using (19)

( ) ( ) ( ) ( ) ( ) ( ) ( )0 11 0 0 0 01 1

2 2

T T

m

A A

U t t t t dA t t t dA = D D ! , (23)

( ) ( ) ( ) ( ) ( ) ( ) ( )22 0 01 1

2 2

T T

s

A A

U t t t t dA t t t dA = " D " " D M , (24)

gde su: where

( ) ( ) ( ) ( ) ( )311 22and /12t t t t t t = =D D D D . (25)

Trenutne deformacije su elasti#ne, pa imamo: The instantaneous strains are elastic, so we have:

( ) ( ) ( )

( ) ( ) ( )

0 11 0 0 0

0 22 0 0

,

,

t t t

t t t

=

=

! D

M D " (26)

gde su: where:

( ) ( )11 0 11 22 0 22andt t= =D D D D . (27)

Koriste%i izraze (26) za promene energija defor-macije, dobijamo:

Using the equations (26) for the strain energyalterations, we get

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

0 11 0 0 0 11 0 0 0

22 0 22 0 0

1 1,

2 2

1 1.

2 2

T T

m

A A

T T

s

A A

U t t t t dA t t t t dA

U t t t t dA t t t t dA

=

=

D D D

" D " D " D " (28)

Rad spoljanjih sila koje izazivaju te#enje piemo uobliku:

The works of external forces that cause the creepare written as:

( ) ( )

( ) ( )

,

.

T

u u u

T

w w w

t t

t t

=

=

W q Q

W q Q

(29)

Ukupne promene potencijalnih energija su sada: The total potential energy alterations are now:


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9

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( )

1 11 1 0 1 11 0 1 0

3 22 3 0 3 22 0 3 0

1 1,

2 2

1 1.

2 2

T T T T T

m u u u u u u u u u u

A A

T T T T T

s w w w w w w w w w w

A A

t t t t dA t t t t dA t

t t t t dA t t t t dA t

=

=

q B D B q D q B D B q q Q

q B D B q D q B D B q q Q

(30)

Primenom stava o minimumu ukupnih promenapotencijalnih energija, diferenciraju%i (30) po vektorima

( )T

u tq i ( )

T

w tq dolazimo do slede%ih jedna#ina

ravnotee:

Minimizing with respect to the vectors ( )Tu tq and

( )Tw tq , leads to the following equations of balance:

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

0 0 0

0 0 0

,

.

uu u uu u u

ww w ww w w

t t t t t

t t t t t

= +

= +

K q D K q Q

K q D K q Q (31)

U jedna#inama (31) podrazumevaju%i aproksimacijupo metodu kona#nih traka [7] jesu:

Assuming the approximation by the FSM accordingto [7], in the equations (31) the following is valid:

( ) ( )

( ) ( )

( ) ( )

( ) ( )

1 11 1

0 1 11 0 1

3 22 3

0 3 22 0 3

,

,

,

,

,

.

T

uumn u m u n

A

T

uumn u m u nA

T

um um u

A

T

wwmn w m w n

A

T

wwmn w m w n

A

T

wm wm w

A

t t dA

t t dA

dA

t t dA

t t dA

dA

=

=

=

=

=

=

K B D B

K B D B

Q A p

K B D B

K B D B

Q A p

(32)

Kako su trenutne deformacije elasti#ne, imamo: Since the instantaneous strains are elastic, we have

( ) ( )

( ) ( )

00 0

0

0 0

,

.

uumn um um

wwmn wm wm

t t

t t

==

K q Q

K q Q (33)

Jedna#ine ravnotee (33) jesu linearne jedna#ine u

vremenu0

t .

Jedna#ine ravnotee (31) sada su:

These two are equations of balance of linear

elasticity at time0

t . The equations of balance (31) are

now:

( ) ( ) ( )

( ) ( ) ( )

0

0

0

0

,

.

uumn um um um

wwmn wm wm wm

t t t

t t t

= +

= +

K q D Q Q

K q D Q Q

(34)

i predstavljaju generalizaciju linearnih jedna#inaravnotee.

Prvo se odre,uje trenutno elasti#no reenje u

trenutku optere%enja0

t . Zatim se za odabrano vreme t

prora#unavaju koeficijenti te#enja, starenja i na osnovu

njih ( )1K t i ( )2K t , te se zadatak ponovo reava.Postupak prora#una mogu%e je sprovesti za bilo koju

algebarsku vezu napona i deformacija.

2.4 Ra*unarski program i interfejs za metodkona*nih traka

Jedna#ine (34) izvedene u ta#ki 2.3 jesu

They represent the generalisation of the equations ofbalance of linear elasticity.

First we determine the instantaneous elastic solution

at the moment of loading0

t . Then we calculate the

coefficients of creep, ageing, and based upon these,

( )1K t and ( )2K t , for selected time t . Then we solve

the problem again.This procedure of the calculation can be performed

for any algebraic relationship of strain and stress.

2.4 Finite strip computer program and interface

The equations (34) which were mentioned in section


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10

generalizacija jedna#ina linearne teorije elasti#nosti, akao rezultat dobijaju se deformacijske i naponskeveli#ine, u zavisnosti od vremena. Ovo omogu%ava dase ve% opisani program [7] za reavanje linearnoelasti#nih problema proiri i da se pomo%u njegaomogu%i prvo reavanje trenutnog t0, linearno elasti#nogproblema, a zatim da se za proizvoljni trenutak vremena

t i sra#unate ( )1K t i ( )2K t , zadatak jo jednom

reava.Zbog toga i organizacija programa za linearno

viskoelasti#nu analizu koritenjem kona#ne trake prikazane na slici 1, omogu%uje da ubacimo petlju povremenu t u linearno elasti#an ra#unarski programispred petlje po #lanovima reda. U prvom koraku,reenje je u vremenu t0 , dok je u drugom to reenje uproizvoljnom vremenu t.

2.3 are generalization of equations of linear theory ofelasticity. These equations provide time-dependentstress and strain values. This allows extending thedescribed program [7] for solving linear elastic problems,

so that it enables us first to solve the instantaneous0t

of the linear elastic problem, and then to solve theproblem once more for the arbitrary time t and already

calculated ( )1K t and ( )2K t .Owing to this, organization of the program for linearviscoelastic analysis using the finite strip shown in Fig. 1provides that we can insert the loop of time t in linearelastic computer program in front of the loop of seriesterms. In the first step it is a solution in the time

0t , while

in the second it is a solution in an arbitrary time t.

Sl. 1. Mrea od dvije kona)ne trake plo)astog nosa)a gdje je jedna traka kabal za prethodno naprezanjeFigure 1. Two finite strips net of plate girder where one strip is prestressing tendon

Prora#un karakteristika kona#nih traka za razli#itavremena omogu%en je interfejsom ra#unarskogprograma. Interfejs programa Metod kona#nih traka uviskoelasti#nim problemima konstrukcija realizovan jeprogramskim jezikom Delphi u razvojnom okruenjuCodeGear RAD Studio 2009. Pri realizaciji interfejsa,kori%ene su standardne biblioteke razvojnog okruenjakoje su bile neophodne, uz napomenu da je grafi#ki

prikaz kona#nih traka ura,en pomo%u TeeChartbiblioteke klasa (ver. 8). Radi bolje funkcionalnostiinterfejsa, kori%ene su i Developer Expresskomponente (ver. 44) i njihove biblioteke klasa koje nedolaze s pomenutim razvojnim okruenjem, ve% sunaknadno uklju#ene.

Interfejs omogu%ava unos svih potrebnih parametaraza izvravanje MKTVE algoritma, kao i real-timegrafi#kiprikaz kona#nih traka. Nakon zavretka unosa, mogu%ee generisanje datoteke neophodne za izvravanjealgoritma i poziv pomenutog algoritma. Izvravanjemalgoritma, kreira se datoteka u kojoj su izlazni rezultati.Ulazna i izlazna datoteka #uvaju se u direktorijumu, gdese nalazi i sam MKTVE algoritam.

The calculation of characteristics of finite strips fordifferent times is enabled by the interface of thecomputer program. The interface of the "Finite stripmethod in viscoelastic structural problems" program wasimplemented in the Delphi programming language usingCodeGear RAD Studio 2009 development environment.In the implementation of interfaces, all the standardlibraries of the development environment that were

necessary were used, noting that the graphicalrepresentation of the finite strips was made usingTeeChart class library (ver. 8). For better functionality ofthe interface the Developer Express components (ver.44) were used, which class libraries do not come withthe aforementioned development environment but weresubsequently included.

Interface allows you to enter all the necessaryparameters to execute the finite strip viscoelastic(FSMVE) algorithm and real-time graphical represen-tation of the finite strips. After completing the input it ispossible to generate a data file required to execute thealgorithm and call the aforementioned algorithm. Theexecution of the algorithm creates a data file in which the


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11

Na slikama datim u nastavku (25), prikazani sudelovi interfejsa gde se radi unos svih parametarapotrebnih za izvravanje MKTVE algoritma.

output results are contained. Input and output data fileare saved in the directory where the FSMVE algorithm isplaced.

On figures 2 - 5, which are given below, the parts ofthe interface are shown, where all the parametersneeded to execute FSMVE algorithm are entered.

Sl. 2. Unos osnovnih parametaraFigure 2. The input of basic parameters

[

Sl. 3. Unos ta)aka du )vornih linija u kojima se trae momenti i naponiFigure 3. The input of points along the nodal lines in which the moments and stresses are calculated

Sl. 4. Unos broja i koordinata )vornih linijaFigure 4. The total number of nodal lines and their coordinates

Sl. 5. Unos kona)nih traka, pomeranja i optere,enjaFigure 5. The input of finite strips, their displacements and loads


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12

Na slici 6. dat je grafi#ki prikaz kona#nih traka. On figure 6. the graphical representation of the finitestrips is given.

Sl. 6. Grafi)ki prikaz kona)nih trakaFigure 6. Graphical representation of the finite strips

3 NUMERI,KI PRIMERI

3.1 Primer 1 prora*un prethodno napregnutognosa*a I preseka

Za prethodno napregnut nosa# izvreno jedimenzionisanje klasi#nim prora#unom prema teoriji zaprethodno napregnute nosa#e koritenjem literatura[8],[9],[10], koji su onda upore,eni s rezultatima analize

dobijenim kori%enjem metode kona#nih traka programMKTVE.

Na slici 7. date su osnovne dispozicionekarakteristike prethodno napregnutog I nosa#a:

3 EXAMPLES OF COMPUTATION

3.1 Example 1 computation of prestressedconcrete girder with "I" cross-section

This example presents comparative analysis ofcalculations of a prestressed concrete girder accordingto the theory of prestressed concrete using Ref.[8],[9],[10], and to the FSMVE.

In Fig. 7. basic disposition characteristics of a girderwith "I" cross-section are presented:

Sl. 7. Osnovne dispozicione karakteristike nosa)a potrebne za prora)unFigure 7. Basic disposition characteristics of the girder which are required for computation


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13

Predvi,eno je da se nosa#i uteu sa po #etiri kabla12-16, s povrinom popre#nog preseka

2 218 4 72k

F cm cm= = . Debljina trake koja predstavlja

kablove tada je 0.9kk

Ft cm

b= = .

Ukupna sila prethodnog naprezanja je tada:4 2345 9380kN kN= = .

Raspored i detal.i kablova u zoni uvo,enja sileprethodnog naprezanja i u sredini raspona prikazani suna slici 8. i slici 9:

The girder is prestressed with 4 tendons 12-16 with

cross-section area of 2 218 4 72k

F cm cm= = . The finitestrip thickness of tendon is

0.9kk

Ft cm

b= = .

Total prestressing force in the tendons than is4 2345 9380kN kN= = .

Dimensions and reinforcement details of girder arepresented in Fig. 8 and Fig. 9:

Sl. 8. Plan kablova na )elu nosa)a i na sredini rasponaFigure 8. Dimensions and tendon layout on the beginning and in the middle of girder

Sl. 9. Raspored i detal-i kablova izgled C C i presek D DFigure 9. Dimensions and tendon layout view C C and cross-section D D

Radi pojednostavljenja postupka prora#una, zausvojenu trasu kablova uzeta je teina linija kablova.

Karakteristike betona kao viskoelasti#nogmaterijala jesu:

MB 50,2

0 28 3600 /bE E E kN cm= = = ,

kona#na vrednost koeficijenta te#enja 2.5= ,koeficijent starenja betona 0.8= ,

The median line of all tendons in girder is taken asthe tendon route to simplify the computation procedure.

The characteristics of concrete as viscoelasticmaterial are:

C 50,2

0 28 3600 /bE E E kN cm= = = ,

the final value of creep coefficient: 2.5= ,ageing coefficient: 0.8= ,


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15

Napon u betonu za traku sa zadatim karakteri-stikama #elika dobijamo iz izraza:

The stresses in the concrete strips with charac-teristics of tendon are obtained from

( 1)

2

si s i

bin

++= . (37)

Ispitana su dva slu#aja ekvivalentnog optere%enja:Slu#aj A) Ekvivalentno optere%enje ra#unato spadovima sile prethodnog naprezanja za presek nasredini raspona, i slu#aj B) Ekvivalentno optere%enjera#unato s prose#nim padovima sile prethodnognaprezanja.

Ekvivalentno optere%enje ra#unamo prvo zaprora#un po teoriji savijanja grednih nosa#a (TSG). Naosnovu dobijenih padova sile prethodnog naprezanja,prvo odre,ujemo silu prethodnog naprezanja

k,

radijus krivine trase kablova R , te vrednost

ekvivalentnog raspodeljenog optere%enja /k k

q N R=

za vreme 0t t= , i za vreme t t= . Dalje, ekvivalentnooptere%enje ra#unamo za prora#un po metodi kona#nihtraka, koje %emo unositi u program MKTVE.

Two cases of equivalent loads are tested: Case A) Equivalent load calculated with loss of prestressing forcefor the cross-section in the middle of the span lenght,and case B) Equivalent load calculated with theaverage loss of prestressing force.

First we calculate the equivalent load for thecalculation according to the theory of prestressedconcrete (TPC). Based on values of losses ofprestressing forces obtained, we calculate the

prestressing force kN , radius of curvature of median line

of tendons R , and than values of equivalent load

/k k

q N R= mentioned above, at the time0

t t= andat the time t t= . Further, we calculate the equivalentload for the calculation according to the FSMVE.

Sl. 11. Sila prethodnog naprezanja i ekvivalentno optere,enjeFigure 11. Prestressing force and equivalent load

3.1.1 Slu#aj A) Ekvivalentno optere%enje ra#unato spadovima sile prethodnog naprezanja za preseksredini raspona:

3.1.1 Case A) Equivalent load calculated with loss ofprestressing force for the cross-section in themiddle of the span lenght:

Tabela 1. Ekvivalentno optere,enje za prora)un po MKT, za slu)aj A)Table 1. Equivalent load for calculation with FSM, case A)

TSGTPC

MKTVEFSMVE

0t t= t t= 0t t= i t t=

trenjefriction

8,118 %(704,31 kN)

-8,118 %

(704,31 kN)kotvl.enjeanchoring

0 - 0

Padsile

prethodnog

naprezanja

Lossof

prestressing

force

elasti#no skra%enjeelastic strains

4,414 %(382,98 kN)

- -


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16

Ukupno za0

t t= :Total loss at the time

0t t= :

12,532%(1087,29 kN)

12,532%(1087,29 kN)

8,118 %(704,31 kN)

relaksacije #elika,te#enje i skupl.anje betonarelaxation of steel, creepand shrinkage of concrete

-25,800%

(2238, 09kN) -

Ukupno za t t= :

Total loss at the time t t= : -38,332%

(3325,38 kN)

8,118 %

(704,31 kN)

Sila prethodnog naprezanja [ ]kN kN

Prestressing force [ ]k kN 8292,71 6054,62 8675,69

[ ]1 kN 8257,187 6028,684 8638,53Komponente sile kN

Components of kN [ ]1V kN 766,753 559,817 802,163

Radijus krivine trase kablova [ ]R m

Radius of curvature [ ]R m 183,47 183,47 183,47

Ekvivalentno optere%enje kk

N kNq

R m

=

Equivalent load kk

N kNq

R m

=

45,199 33,001 47,287

3.1.2 Slu#aj B) Ekvivalentno optere%enje ra#unato sprose#nim padovima sile prethodnog naprezanja

3.1.2 Case B) Equivalent load calculated with theaverage loss of prestressing force

Tabela 2. Ekvivalentno optere,enje za prora)un po MKT, za slu)aj B)Table 2. Equivalent load for calculation with FSM, case B)

TSGTPC

MKTVEFSMVE

0t t= t t= 0t t= i t t=

trenjefriction

3,531 %(313,833 kN)

- 3,531 %(313,833 kN)

kotvl.enjeanchoring

3,678 %(341,481 kN)

- 3,678 %(341,481 kN)


3,063 %(274,760 kN)

- -

Ukupno za0


0t t= :

10,272%(903,074 kN)

10,272%(903,074 kN)

7,209 %(655,314 kN)

relaksacije #elika,te#enje i skupl.anje betonarelaxation of steel, creep

and shrinkage of concrete

-17,9606%

(1612,16 kN) -

Pro

se#navrednostpadasile

prethodnognaprezanja

Averagelossofprestressing

force

Ukupno za t t= :Total loss at the time t t= :

-28,233%

(2542,24 kN)7,209 %

(655,314 kN)

Sila prethodnog naprezanja [ ]kN kN

Prestressing force [ ]kN kN 8476,926 6837,76 8724,686

[ ]1N kN 8440,613 6808,469 8687,312Komponente sile k

Components of kN [ ]1V kN 783,785 632,226 806,694

Radijus krivine trase kablova [ ]R m 183,47 183,47 183,47


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17

Radius of curvature [ ]R m

Ekvivalentno optere%enje kk

N kNq

R m

=

Equivalent load kk

N kNq

R m

=

46,203 37,269 47,554

3.1.3 Tabelarni i grafi#ki prikaz normalnih napona

U tabelama 3. i 4. prikazane su vrednosti naponadobijene primenom klasi#ne teorije prethodnonapregnutih nosa#a (TSG) i vrednosti napona dobijeneprimenom metode kona#nih traka, program MKTVE, pri#emu je ekvivalentno optere%enje ra#unato s prose#nimpadovima sile prethodnog naprezanja (slu#aj B):

3.1.2 Tabular and graphic representation of normalstresses

Tables 3. and 4. represent values of stress obtainedaccording to the theory of prestressed concrete (TPC)and values of stress obtained according to the finite stripmethod program FSMVE. The values of equivalentload used for these results are computed with averageloss of prestressing force (case B):

Tabela 3. Vrednosti normalnih napona u vremenu0

t t= Table 3. Values of normal stresses at the time

0t t=

prethodno naprezanjeprestressing sopstvena teinaself-weight ukupno optere%enjetotal load#vorna linijaNodal line TSG

TPCMKTVEFSMVE

TSGTPC

MKTVEFSMVE

TSGTPC

MKTVEFSMVE

1 -28.964 -26.667 7.890 3.850 -21.074 -22.817

2 -26.213 -24.129 6.710 3.240 -19.503 -20.889

3 -26.031 -24.044 6.630 3.230 -19.401 -20.814

4 -23.889 -21.987 5.710 2.740 -18.179 -19.247

5 -20.844 -19.182 4.400 2.080 -16.444 -17.102

6 -15.465 -14.217 2.090 0.910 -13.375 -13.307

7 -10.100 -10.479 -0.220 -0.250 -10.320 -10.729

8 -3.793 -3.548 -2.940 -1.610 -6.733 -5.1589 2.500 2.076 -5.640 -2.970 -3.140 -0.894

10 5.545 4.875 -6.940 -3.640 -1.395 1.235

Tabela 4. Vrednosti normalnih napona u vremenu t t= Table 4. Values of normal stresses at the time t t=

prethodno naprezanjeprestressing

sopstvena teinaself-weight

ukupno optere%enjetotal load#vorna linija

Nodal line TSGTPC

MKTVEFSMVE

TSGTPC

MKTVEFSMVE

TSGTPC

MKTVEFSMVE

1 -23.363 -21.249 7.890 3.110 -15.473 -18.1392 -21.145 -19.221 6.710 2.570 -14.435 -16.651

3 -20.997 -19.169 6.630 2.550 -14.367 -16.619

4 -19.270 -18.092 5.710 2.130 -13.560 -15.962

5 -16.814 -15.271 4.400 1.540 -12.414 -13.731

6 -12.475 -11.306 2.090 0.510 -10.385 -10.796

7 -8.147 -8.570 -0.220 -0.510 -8.367 -9.080

8 -3.060 -2.801 -2.940 -1.710 -6.000 -4.511

9 2.017 1.657 -5.640 -2.920 -3.624 -1.263

10 4.473 3.891 -6.940 -3.500 -2.467 0.391


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18

Dobijene vrednosti grafi#ki su predstavljene naslede%im slikama:

In the following Figures, the results of stressescalculated in the middle of the span length are shown.

Sl. 12. Dijagram napona $y[MPa] usled sopstvene teine nosa)a a) prema MKT u vremenu 0t t= ; b) prema MKT uvremenu t t= ; c) prema TSG

Figure 12. Distribution of stress $y[MPa] for self-weight a) according to the FSMVE at the time 0t t= , b) according tothe FSMVE at the time t t= , c) according to the TPC

Sl. 13. Dijagrami napona $y[MPa] usled sile prethodnog naprezanja: a) prema MKT u vremenu 0t t= ; b) prema MKT uvremenu t t= ; c) prema TSG u 0t t= ; d) prema TSG u t t=

Figure 13. Distribution of stress $y[MPa] due to force in the tendon: a) according to the FSMVE at the time 0t t= , b)according to the FSMVE at the time t t= , c) according to the TPC at the time 0t t= , d) according to the TPC at the

time t t=


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Sl. 14. Dijagrami napona $y[MPa] za ukupno optere,enje: a) prema MKT u vremenu 0t t= ; b) prema MKT u vremenut t= ; c) prema TSG u 0t t= ; d) prema TSG u t t= .

Figure 14. Distribution of stress $y[MPa] for total loading: a) according to the FSMVE at the time 0t t= , b) accordingto the FSMVE at the time t t= , c) according to the TPC at the time 0t t= , d) according to the TPC at the time t t=

Padovi sile prethodnog naprezanja, dobijeniprimenom klasi#ne teorije savijanja grednih nosa#a i

primenom metode kona#nih traka za slu#aj A i slu#aj B,upore,eni su u tabeli 5:

Table 5. presents comparison of loss of prestressingforce obtained according to the TPC and according to

the FSMVE for case A) and B):

Tabela 5. Pad sile prethodnog naprezanjaTable 5. Loss of prestressing force

TSGTPC

MKT A)FMSVE A)

MKT B)FMSVE B)

trenjefriction

8,118 %(704,31 kN)

8,118 %(704,31 kN)

8,118 %(704,31 kN)

kotvl.enjeanchoring

0 0 0


4,414 %(382,98 kN)

9,373 %(813,108 kN)

9,4336 %(818,43 kN)

Ukupno za 0t t= :Total loss at the time

0t t= :

12,532%(1087,29 kN)

17,491%(1517,418 kN)

17,5516 %(1522,74 kN)

relaksacije #elika,te#enje i skupl.anje betonarelaxation of steel, creepand shrinkage of concrete

25,800 %(2382,09 kN)

31,8615 %(2763,985 kN)

31,859 %(2763,985 kN)

Padsileprethodnog

naprezanja

Lossofprestress

ingforce


38,332%(3325,38 kN)

39,9795 %(3468,295 kN)

39,977 %(3468,281 kN)


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3.1.4 Diskusija rezultata

Analiziraju%i rezultate dobijene klasi#nom teorijom sa-vijanja grede i primenom metode kona#nih traka, dobija-mo da klasi#na teorija daje manje gubitke sile prethodnognaprezanja u vremenu

0t t= , zato to je u klasi#noj

teoriji gubitak usled elasti#nog skra%enja neto manji.U vremenu t t= gubitak sile prethodnog naprezanja

dobijen klasi#nom teorijom savijanja grede i primenommetode kona#nih traka skoro je izjedna#en, razlika je1,645%, to je zanemarl.ivo. Prema tome, sledi da se uklasi#noj teoriji, poto se zasebno ra#una pad usledelasti#nog skra%enja i pad usled te#enja, skupl.anja betonai relaksacije #elika, dobija dosta ve%i pad usled te#enja,skupl.anja betona i relaksacije #elika nego metodomkona#nih traka, gde su ti padovi povezani.

3.2 Primer 2 prora*un prethodno napregnutognosa*a sandu*astog preseka

Na slici 15. date su osnovne dispozicione

karakteristike prethodno napregnutog nosa#asandu#astog popre#nog preseka:

3.1.4 Results discussion

Analyzing the results obtained from TPC, and to theFSMVE, we find that TPC gives smaller loss ofprestressing force at the time

0t t= , because in TPC the

loss of prestressing force due to elastic strains is less.At the time t t= the loss of prestressing force

obtained by TPC and by FSMVE is nearly equal. Thedifference is 1.645%, which is negligible. Hence, in thetheory of prestressed concrete where the loss ofprestressing force is calculated separately due to elasticstrains and due to creep and shrinkage of concret andrelaxation of steel, we obtain much bigger loss of pres-tressing force due to creep and shrinkage of concreteand relaxation of steel than with the finite strip method -program FSMVE, where those losses are related.

3.2 Example 2 computation of a prestressedgirder with box - section

This example presents comparative analysis of

calculations of a prestressed concrete girder accordingto the theory of prestressed concrete, and to the FSMVE.Figure 15. provides basic dispositional charac-

teristics of prestressed girder with box cross-section:

Sl. 15. Osnovne dispozicione karakteristike nosa)a potrebne za prora)unFigure 15. Basic disposition characteristics of the girder which are required for computation


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Predvi,eno je da se nosa#utee sa po est kablova6-15.2, s po#etnom silom prethodnog naprezanja N0=6204 kN. Me,utim, radi pojednostavljenja postupkaprora#una, za usvojenu trasu kablova uzeta je teinalinija kablova, koja du nosa#a ima oblik polinoma tre%egreda.

Sa e ozna#eno je rastojanje teine linije kablova i

teita popre#nog preseka (slika 16):

It is provided that the girder is prestressed with 6tendons 6-15.2, with the initial prestressing force N0=6204 kN. However, in order to simplify the calculationprocedure, for the applied cable route was taken medianline of the tendons, which along the girder has a form ofa polynomial of the third order.

With an e is marked distance line cables gravity

and cross section gravity (Fig. 16):

Sl. 16. Teina linija kablova za polovinu nosa)aFigure 16. Median line of cables for half length of girder

Karakteristike betona - kao viskoelasti#nogmaterijala jesu:

20 28 3400 /bE E E kN cm= = = ,

koeficijent te#enja 0( , ) 2.552t t = ,

koeficijent starenja 0

( , ) 0.75t t = ,odakle su parametri:

01

28

1 3.552E

kE

= + = ,

2 1 2.914k = + = .Modul elasti#nosti prethodno napregnutih kablova

jeste219500 /

kE kN cm= ,

dok je Poisson-ov koeficijent0= .Odavde sledi da je odnos modula elasti#nosti #elika i

betona

5.74k

b

En

E= = .

Na slici 17. prikazan je raspored #vornih linija ikona#nih traka unutar popre#nog preseka u usvojenomkoordinatnom sistemu. Pri prora#unu je uzeta u obzirsimetrija popre#nog preseka. Koritena je kona#na trakaplo#astog nosa#a s #etiri stepena slobode po #vornojliniji, kao i u prethodnom primeru. Popre#ni presekdiskretizovan je sa deset kona#nih traka.

The characteristics of concrete as viscoelasticmaterial are:

20 28 3400 /bE E E kN cm= = = ,

the final value of creep coefficient:

0( , ) 2.552t t = ,ageing coefficient:

0( , ) 0.75t t = ,

and the following parameters yield:

01

28

1 3.552E

kE

= + = ,

2 1 2.914k = + = .The modulus of elasticity and Poissons coefficient of

prestressing tendon is:2

19500 /kE kN cm= ,

0= ,and following ratio of modulus of elasticity of steel andconcrete yields:

5.74k

b

En

E= = .

Figure 17. presents the distribution of nodal pointsand the finite strip mesh within the cross section. Thecalculation takes into account the symmetry of the cross-section. Flat shell finite strip with four degrees offreedom per nodal line was used as in the previousexample. Cross-section is discretized with 10 finitestrips.


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23

Sl. 18. Sila prethodnog naprezanja i ekvivalentno optere,enjeFigure 18. Prestressing force and equivalent load

Tabela 7. Vrednost pada sile prethodnog naprezanja, sila prethodnog naprezanja i ekvivalentnih optere,enja za MKTVE,za slu)aj A)

Table 7. The value of loss of prestressing force, prestressing force and equivalent load for FSMVE, case A)

MKTVE

FSMVE

0t t= t t=

trenjefriction

2,5% 2,5%

kotvl.enjeanchoring

0 0


3,91% -

Ukupno za 0t t= :Total loss at the time

0t t= :

6,41% -

elasti#no skra%enje sa skupljanjem i te#enjem betona i

relaksacijom #elikaelastic strains with relaxation of steel, creep and shrinkage ofconcrete

- 20,67%

Padsile

prethodnognaprezanja

Lossofprestressingforce


- 23,17%

[ ]1N kN 6052,62 6052,62Komponente sile kN

Components ofk

N [ ]1V kN 508,42 508,42

Ekvivalentno optere%enjek

kNq

m

Equivalent loadk

kNq

m

31,94 31,94

3.2.2 Slu#aj B) Ekvivalentno optere%enje ra#unato sprose#nim padovima sile prethodnog naprezanja

Za prora#un ekvivalentnog optere%enja ukupna silaprethodnog naprezanja umanjena je za prose#nuvrednost pada sile prethodnog naprezanja usled trenja ikotvljenja:

3.2.2 Case B) Equivalent load calculated with theaverage loss of prestressing force

For the calculation of the equivalent load, totalprestressing force is lower for the value of average lossof prestressing force caused by friction and anchoring:


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Tabela 8. Prose)na vrednost pada prethodnog naprezanja, sile prethodnog naprezanja i ekvivalentna optere,enja zaMKTVE, za slu)aj B)

Table 8. The value of average loss of prestressing force, prestressing force and equivalent load for FSMVE, case B)

MKTVEFSMVE

0t t= t t= trenjefriction

2,30% 2,30%

kotvl.enjeanchoring

0,99% 0,99%


3,85% -

Ukupno za0


0t t= :

7,14% -

elasti#no skra%enje sa skupljanjem i te#enjem betona irelaksacijom #elikaelastic strains with relaxation of steel, creep and shrinkage of

concrete

- 20,72%

Pad

sileprethodnognaprezanja

L

ossofprestressingforce


- 24,01%

[ ]1 kN 6005,66 6005,66Komponente sile kN

Components ofk

N [ ]1V kN 504,47 504,47

Ekvivalentno optere%enjek

kNq

m

Equivalent loadk

kNq

m

31,69 31,69

Padovi sile prethodnog naprezanja, dobijeni prime-nom klasi#ne teorije savijanja grednih nosa#a i prime-nom metode kona#nih traka za slu#aj A i slu#aj B,upore,eni su u tabeli 9:

The loss of prestressing force obtained by TPC andFSMVE for Case A and Case B are compared in Table9:

Tabela 9. Pad sile prethodnog naprezanjaTable 9. Loss of prestressing force

TSGTPC

MKTVE A)FSMVE A)

MKTVE B)FSMVE B)

Pad sile u0

t t=

Loss of prestressing force at the time 0t t= 7,17 % 6,41 % 7,14 %

Pad sile u t t= Loss of prestressing force at the time t t=

21,49 % 23,17 % 24,01 %

3.2.3 Tabelarni i grafi#ki prikaz normalnih napona

Analizirani rezultati grafi#ki predstavljeni su nanarednim slikama:

3.2.3 Tabular and graphical presentation of normalstresses

Obtained results are shown on the next figures:


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Sl. 19. Dijagrami napona $y[MPa] usled sile prethodnog naprezanja 1) rezultati po TSG slu)aj A); 2) rezultati poMKT slu)aj A); 3) rezultati po MKT slu)aj B)

Figure 19. Distribution of stress $y[MPa] due to force in the tendon: 1) according to the TPC case A); 2) according tothe FSMVE case A); 3) according to the FSMVE case B

U tabelama 10. i 11. prikazane su vrednosti napona

1y [MPa] usled prethodnog naprezanja, sopstvene tei-ne i ukupnog optere%enja, date za vreme

0t t= i

t t= :

Tables 10 and 11 shows stress values 1y [MPa]due to prestressing, self-weight and total load, given at

the time0

t t= and at the time t t= :

Tabela 10. Vrednosti normalnih napona u vremenu0

t t= Table 10. Values of normal stresses at the time 0t t= :




Nodal line TSG

TPC

MKTVE

FSMVE

TSG

TPC

MKTVE

FSMVE

TSG

TPC

MKTVE

FSMVE1 -2.61 -2.41 1.72 1.58 -0.89 -0.832 -2.61 -2.41 1.72 1.58 -0.89 -0.833 -2.61 -1.93 1.72 1.11 -0.89 -0.824 -2.07 -1.55 1.21 0.73 -0.86 -0.825 -1.55 -1.15 0.71 -1.34 -0.84 -2.496 -1.02 -0.63 0.20 -0.11 -0.82 -0.747 -0.49 -0.36 -0.30 -0.44 -0.79 -0.808 0.03 0.05 -0.81 -0.86 -0.77 -0.819 0.69 0.39 -1.31 -1.19 -0.62 -0.80

10 0.80 0.39 -1.62 -1.18 -0.82 -0.7911 0.80 0.39 -1.62 -1.16 -0.82 -0.77


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26

Tabela 11. Vrednosti normalnih napona u vremenu t t= Table 11. Values of normal stresses at the time t t= :




Nodal line TSG

TPC

MKTVE

FSMVE

TSG

TPC

MKTVE

FSMVE

TSG

TPC

MKTVE

FSMVE1 -2.28 -1.52 2.22 1.26 -0.06 -0.262 -2.28 -1.61 2.22 1.12 -0.06 -0.493 -2.28 -1.30 2.22 0.74 -0.06 -0.564 -1.76 -1.05 1.56 0.44 -0.20 -0.615 -1.29 -0.78 0.91 0.12 -0.39 -0.666 -0.83 -0.41 0.25 -0.24 -0.58 -0.647 -0.36 -0.25 -0.40 -0.50 -0.76 -0.758 0.10 0.02 -1.06 -0.84 -0.95 -0.819 0.57 0.25 -1.71 -1.10 -1.14 -0.85

10 0.70 0.26 -2.09 -1.09 -1.39 -0.8411 0.70 0.26 -2.09 -1.08 -1.39 -0.82

Raspored napona za prethodno naprezanje,sopstvenu teinu i ukupno optere%enje dat je naslede%im slikama:

Stress distribution for prestressing, self-weight andtotal load are shown on next figure:

Sl. 20. Dijagrami napona $y[MPa] za slu)aj A: 1) usled sile prethodno naprezanja; 2) usled sopstvene teine; 3) zaukupno optere,enje; a) prema MKT u vremenu

0t t= ; b) prema MKT u vremenu t t= ; c) prema TSG u vremenu

0t t=

Figure 20. Distribution of stress $y[MPa] for case A : 1) due to prestressing force; 2) due to self-weight; 3) for total load,a) according to the FSMVE at the time

0t t= , b) according to the FSMVE at the time t t= , c) according to the TPC at

the time0

t t=


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3.2.4 Diskusija rezultata

Pad sile prethodnog naprezanja, sra!unat metodomkona!nih traka u vreme t = t0, odgovara padu koji se uTeoriji prethodno napregnutog betona naziva pad sileprethodnog naprezanja usled elasti!nih deformacijabetona. Da bismo u prora!unu imali u vidu i ostalepadove sile prethodnog naprezanja (trenje i kotvljenje),njih smo zadali u sklopu ekvivalentnog optere$enja, i totako to smo umanjili po!etnu silu prethodnognaprezanja za te padove. Upore&ivanjem rezultata,moemo zaklju!iti da pad sile prethodnog naprezanjadobijen Klasi!nom teorijom savijanja greda (7.17%)odgovara padu sile prethodnog naprezanja dobijenommetodom kona!nih traka za slu!aj B), odnosno kada je uprora!un unesen prose!an pad usled trenja i kotvljenjakablova. (7.14%). Za slu!aj A), odnosno kada je uprora!unu uzet u obzir pad sile prethodnog naprezanjausled trenja i kotvljenja za presek na sredini raspona,metod kona!nih traka daje manji pad sile u vremenu t =t0(6.41%).

Trajna sila prethodnog naprezanja, dobijenaKlasi!nom teorijom savijanja greda, za 21.49% je manja

od po!etne sile, dok je ukupan pad sile prethodnognaprezanja, dobijen metodom kona!nih traka 23.17%(za slu!aj A) ), odnosno 24.01% ( za slu!aj B) ).

Naposletku, moemo zaklju!iti da program MKTVE,u slu!aju ovog primera, daje ve$i pad sile prethodnognaprezanja nego Klasi!na teorija savijanja greda razlika je ~ 2.00 %.

3.2.4 Results discussion

Loss of prestressing force calculated with the finitestrip method program FSMVE at the time t = t0corresponds to that which is known in TPC as "loss ofprestressing force caused by elastic strains of concrete".In order to take into account other losses of prestressingforce (friction and anchoring), we have struck them aspart of an equivalent load, and so we reduce the initialprestressing force for that loss. Comparing the results,we conclude that loss of prestressing force, obtained bythe CPT (7.17%) corresponds to a loss of prestressingforce obtained by FSMVE computer program for caseB), when we included in calculation the average loss ofprestressing force caused by friction and anchoring ofthe tendons (7.14%). In case A) when in calculation isconsidered loss of prestressing force caused by frictionand anchoring for cross-section, the computer programFSMVE gives a lower loss of prestressing force at thetime t = t0(6.41%).

Permanent prestressing force, obtained by CPT is for21.49 % less than the initial force, while the overall lossof prestressing force obtained by FSMVE computer

program is 23.17 % (in case A) ), and 24.01 % ( in caseB) ).

Finally we can conclude that, in the case of thisexample, the finite strip method program FSMVEprovides a greater loss of prestressing force than TPC,the difference is ~ 2.00 %.

4 ZAKLJUAK

Viskoelasti!ni metod kona!nih traka prezentovan je uprora!unu prethodno napregnutih betonskih grednihnosa!a. Metod kona!nih traka pretstavlja kombinaciju

ravnog stanja napona i Kirchhoff-ve teorije savijanjaplo!a gdje su matrice krutosti i vektori optere$enjaizraeni analiti!ki kao funkcija amplituda !vornihpomeranja. Kompleksni matemati!ki izrazi programiranisu u okviru ra!unarskog jezika Fortran. Interfejsprograma realizovan je programskim jezikom Delphi urazvojnom okruenju CodeGear RAD Studio 2009.Imaju$i u vidu da je metod kona!nih traka poluanaliti!ki ipolunumeri!ki metod, u teoriji poliedarskih ljuski, on jeveoma povoljan za reavanje problema plo!astih nosa!au pore&enju s prora!unom prema teoriji prethodnonapregnutog betona.

4 CONCLUSION

The viscoelastic finite strip method has beenpresented for use in the design of prestressed concretegirders. Finite strip method represents a combination of

plane stress and Kirchhoffs theory of bending plateswhere the stiffness matrices and loading vectors areexpressed analytically as a function of the nodaldisplacement amplitudes. Complex mathematicalexpressions were programmed within the frame of theFORTRAN programming language. The interface of theprogram was implemented in the Delphi programminglanguage using Code Gear RAD Studio 2009development environment. Bearing in mind that the finitestrip method is a semi-analytical and semi-numericalmethod, in the theory of polyhedral shells, we find it veryfavourable in solving of the folded plate structuresproblem in comparison with the calculation according tothe theory of prestressed concrete.


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5 LITERATURAREFERENCES

[1] Timoenko, S. and Gudier, J.H.: Theory ofelasticity, GK (translate from English intoSerbo-Croatian), Beograd, 1962.

[2] Reiner, M.: Rhologie thorique, Dunod, Paris,1955.

[3] Baant, Z.P. and Kim, J.-K.: Improved predictionmodel for time-dependent Deformations ofconcrete: Part 5-Cyclic load and cyclic humidity,Materials and Structures 25, 1992, 163-169.

[4] Hirst, G. A. and Neville, A.M.: Activation energy ofcreep of concrete under short-term static and cyclicstresses, Magazin of concrete research 29, 1998,13-18.

[5] Milainovi%, D. D.: Rheological-dynamical analogy:modeling of fatigue behaviour, International Journalof Solids and Structures 40(1), 2003, 181-217.

[6] [Cumbo, A. and Foli%, R.: Prilog analizi uticajate#enja i skupljanja betona kod spregnutihkonstrukcija primenom metode kona#nihelemenata (in Serbian), Gra,evinski materijali ikonstrukcije 43(1-2), 2000, 12-19.

[7] Milainovi%, D. D.: The Finite Strip Method inComputational Mechanics, Faculties of Civil

Engineering: University of Novi Sad, TechnicalUniversity of Budapest and University of Belgrade:Subotica, Budapest, Belgrade, 1997.

[8] Alendar, V. and Perii%, .: Prethodno napregnutibeton primeri za vebe, Gra,evinski fakultetUniverziteta u Beogradu, 1984.

[9] Pejovi%, R.: Prethodno napregnuti beton,Univerzitet Crne Gore, 1999.

[10] Jefti%, D.: Prednapregnuti beton, Gra,evinskaknjiga, Beograd,1986.

REZIME

VISKOELASTI,NA ANALIZA PRETHODNONAPREGNUTIH BETONSKIH GREDNIH NOSA,AMETODOM KONA,NIH TRAKA

Dragan D. MILAINOVIMila SVILARNataa MR#ALaura TUZABranislav NOVKOVLjubomir MILAINOVI

Metod kona#nih traka uspeno je uveden u studijuviskoelasti#nog ponaanja prethodno napregnutihbetonskih grednih nosa#a. Razlog za uvo,enje ovogmetoda jeste to to je on u reavanju nekoliko tipovaprakti#nih problema mnogo bri pri reavanju od mnogoobuhvatnijeg i prilagodljivijeg metoda kona#nihelemenata. Ovo generalno vai za konstrukcije spravilnom geometrijom i jednostavnim grani#nimuslovima, #ija je diskretizacija u mnogo kona#nihelemenata #esto vrlo skupa. U takvim slu#ajevima,metod kona#nih traka moe biti veoma konkurentan upogledu trokova i ta#nosti, kako tokom prora#una, takoi u prakti#noj primeni. Diskretizacija popre#nog presekau mreu od samo nekoliko kona#nih traka omogu%ava

reavanje sloenih konstrukcijskih problema.Klju*ne re*i: metod kona#nih traka; viskoelasti#na

analiza; prethodno napregnuti betonski gredni nosa#i.

SUMMARY

VISCOELASTIC ANALYSIS OF PRESTRESSEDCONCRETE GIRDERS BY THE FINITE STRIPMETHOD

Dragan D. MILASINOVICMila SVILARNataa MRDJALaura TUZABranislav NOVKOVLjubomir MILASINOVIC

The finite strip method has successfully introduced inthe study of viscoelastic behaviour of prestressedconcrete girders. The reason for the introduction of thismethod lies in the fact that resolving of several classesof practical problems it is much faster than the morecomprehensive and adaptable finite element method.This is generally valid for structures with regulargeometrical shape and simple boundary conditions,whose discretization into many finite elements is oftenvery expensive. In such cases the finite strip method canbe extremely competitive in terms of cost and accuracy,both during computations and in practical application.Discretization of the cross-section into a mesh of onlyseveral finite strips enables the solution of complex

structural problems.Keywords:Finite strip method, viscoelastic analysis,

prestressed concrete girders


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UTICAJ DODATKA RECIKLIRANE GUME I RECIKLIRANOG STAKLA NA PROMENUBRZINE ULTRAZVUKA U BETONU

EFFECTS OF ADDITION OF RECYCLED RUBBER AND RECYCLED GLASS ON THEVARIATIONS OF ULTRASONIC VELOCITY IN CONCRETE

Duan GRDINenad RISTIGordana TOPLI/IUR/I

ORIGINALNI NAU NI RADORIGINAL SCIENTIFIC PAPERUDK: 666.972.16:628.477.043 = 861

1 UVOD

Tehnolozi betona ve% decenijama su zainteresovaniza istraivanje svojstava betona nedestruktivnimmetodama.Tako, na primer, primena vibracionih metodau istraivanju svojstava betona datira s po#etkatridesetih godina prolog veka. Pionirima u toj oblastismatraju se Pauers [11], Obert [9], Hornbruk [4] iTomson [18] (citirano prema [8]). Od tada do danas,razvijene su brojne nedestruktivne metode i mnogiodgovaraju%i merni instrumenti.

Metoda ultrazvuka uspeno se koristi za procenukvaliteta betona vie od ezdeset godina [8]. Razvojultrazvu#ne metode po#eo je najpre u Kanadi i uEngleskoj, skoro u isto vreme, posmatrano s dananjevremenske distance. Kanadski istraiva#i Lesli i izmenrazvili su instrument soniskop [7], dok je Dons uEngleskoj razvio instrument koji je nazvao ultrazvu#niaparat koji se samo u nekim detaljima razlikovao odkanadskog aparata [5] (citirano prema [8]). Nakonizvesnog perioda razvoja ultrazvu#nih aparata i sprove-denih eksperimenata u laboratorijskim uslovima, ezde-setih godina XX veka, ultrazvu#na metoda po#inje da sekoristi i na terenu [8]. Ubrzo je metoda ultrazvuka za

ispitivanje betona standardizovana i prihva%ena u mno-gim zemljama. U Srbiji je to u#injeno preko standardaSRPS U.M1.042:1998 i SRPS EN 12504-4:2008. Autoriiz Srbije su tako,e radili istraivanja u oblasti primene

Duan Grdi%, MSc, student doktorskih studija, Gra,evinsko-arhitektonski fakultet Univerziteta u Niu,[email protected] Risti%, asistent, Gra,evinsko-arhitektonski fakultetUniverziteta u Niu, [email protected] Topli#i%-2ur#i%, docent, Gra,evinsko-arhitektonskifakultet Univerziteta u Niu,[email protected]

1 INTRODUCTION

Concrete technologists have been interested inresearch of the concrete properties for decades. Thus,for instance, implementation of vibration methods inresearch of concrete properties dates back to the early30s of the previous century. Those who are consideredthe pioneers in this field are Powers [11], Obert [9],Hornibrook [4] and Thomson [18], cited by [8]. Sincethen a large number of non-destructive methods andcorresponding measuring instruments have beendeveloped.

The ultrasonic method has been successfully usedfor evaluation of concrete quality for more than 60 years[8]. The development of ultrasonic method started at firstin Canada and England, almost at the same time, fromtodays perspective. The Canadian researchers, Leslieand Cheesman developed the soniscope instrument[7], while in England, Jones developed an instrument hecalled the ultrasonic device, which was different from theCanadian apparatus only in some details [5], cited by [8].After a certain period of development of ultrasounddevices and experiments conducted in laboratoryconditions, at some point in the 60s the ultrasound

method started to be used in the field [8]. Very soon, theultrasound method for concrete testing was standardizedand accepted in a large number of countries. In Serbia,it was accomplished through the standards SRPS

Dusan Grdic, MSc, University of Nis, Faculty of CivilEngineering and Architecture, PhD student,[email protected] Ristic, University of Nis, Faculty of Civil Engineeringand Architecture, assistant lecturer,[email protected] Toplicic-Curcic, University of Nis, Faculty of CivilEngineering and Architecture, ,ssistant professor,[email protected]
mailto:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]:[email protected]


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Tabela 1. Korekcija brzine ultrazvuka zbog promene temperatureTable 1. Ultrasound velocity correction due to the temperature variation

KorekcijaCorrection

[%]Temperatura betona

Concrete temperature[3C] Beton osuen na vazduhu

Concrete dried in airVodom zasi%en beton

Concrete saturated with water

60 + 5 + 440 + 2 + 1,720 0 00 - 0,5 - 1

< -4 - 1,5 - 7,5

Moe se re%i da brzina ultrazvuka ne zavisi odnaponskog stanja u betonskom elementu. Nekosmanjenje brzine ultrazvuka moe se desiti ukoliko jebeton izloen znatnom stati#kom optere%enju, na primer,od 60% do 70% ili vie u odnosu na njegovu #vrsto%u pripritisku. Ovo se objanjava pojavom mikroprslina unutarstrukture betona.

Postojanje #eli#ne armature u betonu smatra seednim od najzna#ajnijih faktora koji mogu uticati nabrzinu ultrazvuka [14]. Brzina ultrazvuka je od 50% do70% ve%a u #eliku nego u betonu. Shodno tome, #estose moe o#ekivati ve%a brzina ultrazvuka u armiranombetonu nego u onom bez armature. Stoga, preporu#ujese da se za merenja odaberu mesta na kojima searmatura ne prua u pravcu puta koji prelazi ultrazvuk.Ukoliko to nije mogu%e, onda se prilikom izra#unavanjabrzine ultrazvuka moraju koristiti odgovaraju%i koeficijentiza korekciju (slika 1 [12]). Pored svega navedenog,postoji zna#ajna verovatno%a da dobijeni rezultatimerenja ne budu dovoljno ta#ni.

It can be stated that the ultrasonic velocity does notdepend on the stress state in a concrete element. Acertain reduction of ultrasonic velocity can occur if theconcrete is exposed to a considerable static load, forinstance of 60 to 70% or more in respect to itscompressive strength. This is explained by theemergence of micro-cracks inside the concrete structure.

The presence of steel reinforcement is consideredone of the most important factors which can affect theultrasonic velocity [14]. The ultrasonic velocity is foraround 50 to 70% higher in steel in comparison toconcrete. Accordingly, often a higher velocity ofultrasound can be expected in the reinforced concretethan in that with no reinforcement. Therefore it isconsidered that when performing measuring, themeasuring points are those where there is noreinforcement lying in the direction of the path travelledby the ultrasound. If this is not possible, then appropriatecorrection coefficients must be used when calculatingultrasound velocity, figure 1 [12]. Even with all thoseprecautions there is a significant possibility that themeasuring results are not sufficiently accurate.

Slika 1. Uticaj armature na brzinu prostiranja ultrazvuka, armatura paralelna s pravcem prostiranja ultrazvukaFigure 1. Influence of reinforcement on the ultrasound propagation velocity, the reinforcement is parallel to the direction

of ultrasound propagation


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Sastav betona komponentni materijali od kojih seon spravlja tako,e uti#e na brzinu ultrazvuka. To jepotpuno logi#no jer od svojstava komponenti i njihovogudela zavise i svojstva betona. Navedeno se svakakoodnosi i na #vrsto%u pri pritisku, s kojom se brzinaultrazvuka #esto dovodi u korelaciju. S druge strane, nijejednako zna#ajan uticaj svake komponente od kojih sebeton spravlja.

Na primer, uticaj vrste cementa nema zna#ajanefekat na brzinu ultrazvuka za isti dostignuti stepen

hidratacije. Ta#nije, mnogo ve%i uticaj ima dostignutistepen hidratacije cementa, pri #emu je korelacija jasna io#igledna: ve%i stepen hidratacije ve%a brzinaultrazvuka. Pove%anje starosti betona, to je u direktnojkorelaciji s dostignutim stepenom hidratacije, logi#nouti#e na pove%anje brzine ultrazvuka. U po#etku, brzinaultrazvuka raste naglo, dok se kasnije prirast usporava strendom sli#nim kao i u slu#aju prirasta #vrsto%e pripritisku betona tokom vremena.

Hemijski dodaci uti#u na brzinu ultrazvuka akouti#u na stopu hidratacije cementa. U tom smislu, najve%iuticaj imaju regulatori brzine vezivanja i o#vr%avanjabetona, to jest akceleratori i retarderi. Uticaj aditiva tipa

reduktora vode moe se razmatrati preko smanjenogvodocementnog faktora. S druge strane, potpuno jelogi#no da pove%anje vodocementnog faktora uti#e nasmanjenje brzine ultrazvuka. Aeranti nemaju zna#ajnijiuticaj na vezu izme,u brzine ultrazvuka i #vrsto%e pripritisku betona [8], to treba prihvatiti sa rezervom jeraeranti pove%avaju sadraj vazduha u betonu.

Od svih komponentnih materijala, agregat imanajve%i uticaj na brzinu prostiranja ultrazvuka kroz beton.Uticaj agregata moe se sagledati iz aspekta vrste ikoli#ine. Ispitivanjima je utvr,eno da je brzina ultrazvukakroz zrna agregata ve%a nego kroz o#vrslu cementnupastu, to se direktno moe dovesti u vezu s njihovomporozno%u. Ono to je veoma vano jeste to to su

rezultati eksperimentalnih istraivanja pokazali da je zaistu #vrsto%u pri pritisku betona brzina ultrazvuka manjaako je beton spravljen s re#nim agregatom nego ako jespravljen s drobljenim agregatom. Tako,e, postojerazlike u brzini ultrazvuka, u zavisnosti od mineralokogsastava drobljenog agregata [20]. Tako, na primer,brzina ultrazvuka %e biti ve%a u betonu s kre#nja#kimagregatom nego u onom, spravljenom sa andezitom,dijabazom ili bazaltom, za isti nivo #vrsto%e pri pritiskubetona.

Upravo zbog toga to je uticaj agregata na brzinuultrazvuka u betonu veoma zna#ajan, u ovom radu serazmatra problematika brzine ultrazvuka u betonima ukojima je deo mineralnog agregata zamenjenrecikliranom gumom, odnosno recikliranim staklom.

3 EKSPERIMENTALNI DEO

3.1 Materijali kori%eni u eksperimentu

Za spravljanje betonskih meavina kori%en je #istportland cement Holcim CEM I 42,5 R i agregat iz rekeJ. Morave, podeljen u tri frakcije 0/4 mm (u#e%e umeavini 45%), 4/8 mm (25%) i 8/16 mm (30%). Konzi-

The composition of concrete, that is componentmaterials, also affects the ultrasound velocity. This iscompletely logical, because the properties of concretedepend on the properties of its components and theirproportion. This certainly applies to the compressivestrength to which the ultrasound velocity is frequentlycorrelated. On the other hand, the influence of all thecomponents from which the concrete is made is not

equally important.For instance, the cement type has no significanteffect on the ultrasound speed, for the same reachedhydration degree. More accurately, much higherinfluence is affected by the reached cement hydrationdegree, whereby the correlation is clear and obvious the higher hydration degree - the higher the ultrasoundvelocity. The increase of the age of concrete, which isdirectly correlated to the achieved hydration degree,logically results in the increase of ultrasound velocity. Inthe beginning, the ultrasound velocity increases abruptly,while later, the increment slows down with the similartrend as one found in the increase of compressivestrength of concrete in time.

Chemical admixtures also affect the ultrasound

velocity if they affect the cement hydration rate. In theseterms, the highest influence has the concrete bindingand hardening rate regulators, that is accelerants andretarders. The influence of the water reduction typeadmixtures can be analyzed through the reduced water/cement ratio. On the other hand, it is completely logicalthat the increase of water cement ratio has effects onreduction of ultrasound velocity. The air-entrainers haveinsignificant influence on the correlation between theultrasound velocity and compressive strength ofconcrete [8], which must be taken with a reserve, sincethe air-entrainers increase the content of air in concrete.

Of all the component materials, aggregate has themost effect on the propagation of velocity through

concrete. The influence of aggregate can be viewedfrom the aspect of type and of quantity. The testsindicated that the ultrasound velocity through theaggregate grains is higher than through the hardenedcement matrix, which can directly correlated with theirporosity. What is very important is that the results of theexperimental research showed that for the samecompressive strength the ultrasound velocity is lower ifthe concrete is made with the river aggregate incomparison to the concrete made with the crushedaggregate [20]. Thus, for example, the ultrasound velo-city will be higher in the concrete with limestone aggre-gate than with that made with andesite, diabase or basaltfor the same level of compressive strength of concrete.

Exactly for this reason, that the effects of aggregateon velocity of ultrasound in concrete are very important,the issue of ultrasound velocity in concretes where thepo

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