Pouzdanost Ab Montažnih Veza

8/17/2019 Pouzdanost Ab Montažnih Veza

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MATERIJALI I KONSTRUKCIJE 53 (2010) 2 (47-70) 47

POUZDANOST AB MONTAŽNIH VEZA

RELIABILITY OF RC PRECAST JOINT

Damir ZENUNOVI Radomir FOLI

PREGLEDNI R #DUDK: 624.014.7.046:519.213 = 861

1 UVOD

Osnovna funkcija gra'evinskog inžinjerstva jeadekvatno rješenje prihvatanja i provo'enja sila krozsistem (konstrukciju). U tu svrhu se vrši analiza, kojapredstavlja idealizaciju konstrukcije u smisluednostavnijeg, ali logi%nog matemati%kog rješenja, kojesadrži osnovne elemente stvarne konstrukcije. Kodtradicionalnog pristupa analizi, dejstva na konstrukciju sumodelirana kao potpuno definisani, jednostavnigeometrijski ili analiti%ki izrazi. Isto tako svojstvamaterijala su usvojena kao nepromjenljiva unutarkonstrukcije i u vremenu. Naravno, dejstva nagra'evinske konstrukcije nisu nikada u potpunostipoznata u smislu njihovog intenziteta i u%estalostipojave. Usvojene prora%unske veli%ine dejstva suslu%ajne i svaka konstrukcija može biti izložena dejstvuve(eg intenziteta od projektovanog. Savremeni pristupuspostavljanja zavisnosti promjenljivih ulaznih veli%ina iizlaznih veli%ina primjenom vješta%kih neuronskih mrežaprezentiran je u radu [25]. Ovakav pristup je pokazaozadovoljavaju(u preciznost u slu%ajevima kada postojidovoljan broj izmjerenih eksperimentalnih podataka.

Svi gra'evinski materijali sadrže mikrokristalneimperfekcije ili lokalne nedostatke. Može se re(i da jesvaki materijal zbir raznih defekata i za upotrebu sesmatra prihvatljivim ako je zbir defekata predvidljiv. Nepostoje, ni do danas, razvijene teorije koje uspostavljajuodnose izme'u %vrsto(e i deformacije tijela. Tako'e, ne

postoje uokvirena istraživanja fenomena kao što su:plasti%no te%enje i krti lom metala, zamor i puzanje,elasti%ni i postelasti%ni odgovor i sl.

Damir Zenunovi(, Docent Univerziteta u TuzliFakultet za rudarstvo, geodeziju i gra'evinarstvo, Tuzlae-mail:[email protected] Radomir Foli(, Profesor, Departman za gra'evinarstvoFakultet tehi%kih nauka Novi Sade-mail: [email protected]

1 INTRODUCTION

The basic function of building design is adequateacceptance and implementation of loads through thesystem (construction). For this purpose performs theanalysis, which represent of structures idealization interms of simpler, but the logical mathematical solution,which contains the basic elements of the real structures.With the traditional approach to the analysis, the actionson the structure are modeled as fully defined, simplegeometrical or analytical expressions. Also, materialproperties are adopted as fixed within the structure intime. Of course, the actions on civil structures havenever fully known in terms of their intensity andfrequency of occurrence. Adopted design values of theactions are random, and each structure can be exposedto the actions with greater values than design values.Modern approach to establishing the dependence ofvariable input and output values using artificial neuralnetworks is presented in the paper [25]. This approachhas proved sufficiently precise in cases where asufficient number of measured experimental data.

All building materials contain microcrystallineimperfections or local defects. It can be said that eachmaterial is sum of various defects and for the use isconsidered acceptable if the sum of defects arepredictable.Not today, developed theories that relate the strengthand deformation of the body not exist. Also, there are no

framed research of phenomena such as plastic creepand brittle fracture, fatigue, and elastic and post-elasticresponse etc.

Damir Zenunovi(, Assistant Professor, Tuzla University,Faculty of Mining, Geology and Civil Engineering Tuzla,e-mail:[email protected] Radomir Foli(, Professor, Civil Engineering Department,Faculty of Technical Sciences Novi Sad,e-mail: [email protected]

mailto:e-mail:[email protected]:[email protected]:e-mail:[email protected]:[email protected]:[email protected]:e-mail:[email protected]:[email protected]:e-mail:[email protected]


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MATERIJALI I KONSTRUKCIJE 53 (2010) 2 (47-70)48

Na%ini na koje dolazi do otkaza gra'evinskih kon-strukcija, njihova u%estalost, ekonomske i socijalneposljedice, pokazuju prili%nu razliku izme'u hipoteti%kih iirealnih konstrukcija. Prora%unsko optere(enje, uslovi nagradilištu, svojstva materijala, razvijene prora%unskeprocedure i adekvatnost predvi'enih veli%ina i oblikakonstrukcije i njenih elemenata je ponekad daleko odstvarnih parametara. Navedeni parametri su rezultat

kompleksnih me'usobnih odnosa, defekata materijala,strukturalnih odstupanja, ljudskih grešaka, i imaju

karakter slu%ajno promjenljivih. Stoga je za kvantifikacijusposobnosti konstrukcije da ispuni o%ekivane zahtjeveneophodno odre'ivanje mjere uspješnosti, koja senaziva pouzdanost.

Uobi%ajena definicija pouzdanosti je:Pouzdanost je vjerovatno' a da ' e objekat (sistem)

adekvatno ispuniti njegovu zahtijevanu funkciju, udefinisanom vremenskom periodu, pod odre4 enimokolnostima.

Prvi korak u ocjenama konstrukcije korištenjemteorije pouzdanosti jeste pribavljanje svih neophodnihulaznih podataka koji opisuju konstrukciju. Analizapouzdanosti zapo%inje sa identifikacijom razli%itih izvora

rizika na konstrukciji. U radu [21] prezentirano je stabloslu%aja, koje razmatra sve mogu(e uzro%nike otkazaelementa i podsistema koji vodi otkazu sistema. Za svkukomponentu u stablu slu%aja mora se definisati grani%nostanje. Analiza pouzdanosti ukllju%uje prora%un i pred-vi'anje vjerovatno(e narušavanja grani%nog stanja u bilokojem trenutku eksploatacionog vijeka konstrukcije.Vjerovatno(a pojave doga'aja kao što je narušavanjegrani%nog stanja je numeri%ka vrijednost mogu(nostipojave. Iz jedna%ine grani%nog stanja može se za svkiparametar prora%unati vjerovatno(a otkaza pf i indekssigurnosti β. Kada se odredi vjerovatno(a, slijede(i korak je odabir alternative prora%una koja poboljšava pouzda-nost konstrukcije, a minimizira rizik od otkaza. Pri tome

e odgovaraju(e opažanje nepouzdanosti klju%no zasigurnost i efikasnu odluku. Probabilisti%ka prora%unskaprocedura je analiza konstrukcije koja uzima u obzirinformacije probabilisti%kog karaktera o kapacitetu Izahtjevima. Ovakav pristup uslovljava sveobuhvatnuanalizu konstrukcije sa svrhom optimaliziranja zahtjevasigurnosti i ekonomi%nosti.

Koncept pouzdanosti je danas opšte prihva(en zaprojektovanje konstrukcija, iako je dobro poznato da je%esto potrebno zna%ajno pojednostavljenje prora%un-skog problema da bi se ovaj koncept efikasno primijenio.Ovo je prije svega iz slijede(a dva razloga:

(1) U svojoj najjednostavnijoj formulaciji procedurezasnovane na pouzdanosti zahtjevu prikaz performansi

konstrukcije u eksplicitnim relacijama izme'

u varijabli(promjenljivih) dejstava i otpornosti. Ali kada jeponašanje konstrukcije uslovljeno nelinearnostima viševarijabli, kao što je uvijek slu%aj sa betonom, ovakverelacije op(enito su dostupne samo u implicitnom obliku.

(2) Kod konstruktivnih sistema sa više komponenti,potpuna analiza pouzdanosti podrazumjeva analizupouzdanosti dijelova sistema i sistema. Zavisno od brojai rasporeda dijelova ocjena pouzdanosti sistema možepostati vrlo komplikovana pa %ak i prakti%no nemogu(aza velike konstruktivne sisteme.

Ways to come to the failure of civil structures, theirfrequency, economic and social consequences, show areasonable difference between the hypothetical and realstructures. Design actions, site conditions, materialproperties, developed design procedures and theadequacy of the anticipated size and form of structures,and its elements, are sometime far from the realparameters. The mentioned parameters are the result of

complex mutual relations, material defects, structuraldifferences, human errors, and have the character ofrandom variable. Therefore, for the quantification ofstructural capacity to meet expected requirements it isnecessary to define measures of success, which iscalled reliability.

The usual definition of reliability is:Reliability is the probability that the building (system)

is adequate to meet its desired function in the definetime period, under certain circumstances.

The first step in reliability-based assessment is toacquire all necessary input data describing the structure.The reliability analysis starts with the indentification ofthe different sources of risk of the structure. In the paper[21] is presented fault tree, which regards all possible

causal sequences of component and subsystem failuresthat lead to system failure. For all components in a faulttree a limit-state has to be defined. Reliability analysisincludes the calculation and prediction of the probabilityof limit-state violation at any stage during a structure’slife. The probability of the occurrence of an event suchas a limit-state violation is a numerical measure of thechance of its occurring. From the limit state equation canbe calculated for each parameter probability of failure pand safety index β. When the probability determined, thenext goal is to choose design alternatives that improvestructural reliability and minimize the risk of failure.Where appropriate perceptions of uncertainty areessential for safe and efficient decisions. Probabilistic

design procedures is the analysis of structures whichtake into account probabilistic-based information aboutcapacities and demands. This provides a comprehensiveanalysis of structures with the aim of optimizing therequirements of safety and economy.

Reliability-based concepts are nowadays widelyaccepted in structural design, even if it is well knownthat, before such concepts can be effectivelyimplemented, the actual design problem often needs tobe considerably simplified. This is mainly due to the twofollowing reasons:

(1) In their simplest formulation, reliability-basedprocedures require the structural performance to berepresented by explicit functional relationships among

the load and the resistance variables. But, unfortunately,when the structural behavior is affected by severalsources of non-linearity, as always happens for concretestructures, such relationships are generally availableonly in an implicit form.

(2) For structural systems with several components,a complete reliability analysis includes both component-level and system-level estimates. Depending on thenumber and on the arrangement of the components,system reliability evaluations can become verycomplicated and even practically impossible for largestructural systems.


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2 OSNOVE TEORIJE VJEROVATNO%E IMJERODAVNI STATISTI"KI PARAMETRI

Vjerovatno/a odre'enog doga'aja u trenutku vreme-na T se definiše izrazom:

2 BASIS OF THE THEORY OF PROBABILITY ANDTHE RELEVANT STATISTICAL PARAMETERS

The probability of particular events in the moment oftime T is defined expression:

( ) ( ) 0; ≥=≤ t t F t T P (1)

Vrijednost vjerovatno/e može biti u granicama

( ) 10 ≤≤ A P . Prema tome, sigurnost se može definisatikao [ ] 0,1=C P . Isto tako mora vrijediti jednakost:

The value of probability can be inside ( ) 10 ≤≤ A P .

Accordingly, security can be defined as [ ] 0,1=C P . Alsomust be valid equality:

[ ] [ ] [ ] 11 =+⇒=+ failure P success P failure success P (2)

Vjerovatno(a uspješnosti konstrukcije je njenapouzdanost i može se definisati izrazom:

The probability of structure success is its reliabilityand can be defined by expression:

( ) 1=+ f P R (3)

Uvrštavanjem (1) u (3) dobije se funkcijapouzdanosti:

Substituting (1) to (3) obtain the reliability function:

( ) ( ) ( )t T P t F t R >=−= 1 (4)

F(t) je funkcija raspodjele otkaza, koja pokazujevjerovatno(u otkaza sistema do trenutka vremena t. F(t)

se još zove kumulativna funkcija raspodjele. Derivaci-om funkcije raspodjele otkaza dobije se funkcija gustineotkaza,

F (t) is a function of the failure distribution, whichquantify probability of system failure until moment of time

t. The function F (t) is also called cumulative distributionfunction. With derivation of function of failure distributionobtain the function of failure density.

( ) ( )dt

t dF t f = (5)

Prema tome izraz (4) dobija oblik: Thus expression (4) gets the form:

( ) ( ) ( ) ( )∫ ∫ ∞

=−=−=t

t

dt t f dt t f t F t R

0

11 (6)

Promjena intenziteta otkaza u eksploeatacionomvijeku sistema (konstrukcije) može se iskazati pomo(ufunkcije intenziteta otkaza date izrazom [19]:

Change of failure intensity during system (structure)service life can be expressed using a function of failureintensity expressed as [19]:

)(

)()(

t R

t f t =λ (7)

Funkcija pouzdanosti se može izraziti preko funkcijeintenziteta otkaza u obliku:

Reliability function can be expressed through thefunctions of the failure intensity with form:

∫ −=

t

dt t

et R 0)(

)(

λ

(8)

Klju%ni statisti%ki parametri diskretizovane raspodjeleslu%ajno promjenljivih, o%ekivana vrijednost i varijacija,definišu se iz analogije krutog štapa optere(enog nizomvertikalnih koncentri%nih sila f i na udaljenostima x i , kako je pokazano na slici 1.

Iz uslova ravnoteže je poznato da je sila kojauravnotežuje vertikalne sile f i jednaka

Key statistical parameters discrete distribution ofrandom variable, expected value and variation, can bedefined by analogy from solid beam loaded with a seriesof concentric vertical forces f i at distances x i , as shownin Fig. 1.

From the condition of equilibrium is well known thatthe force that balances the vertical forces f i equal to,

∑=

=n

i

i f M

1

(9)

dok se njen položaj odre'uje izrazom, while its position is determined expression,

M

f x

x

n

i

ii∑== 1 (10)

Ako se pretpostavi da su diskretne koncentrisane silesa slike 1 vjerovatno(e mogu(ih doga'aja x 1, x 2 ,...,x n i

If we assume that the discrete concentric force(Fig.1) are probability of possible events x 1, x 2 ,..., x n and


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vrijedi jedna%ina (2) (M=1), izraz (9) se može napisati uobliku:

equation (2) is valid (M = 1), expression (9) can berewritten in the form:

[ ] ∑=

==n

i

ii f x x x E

1

(11)

xx1

f(x)

f 1

x2

f 2

xi

f i

xn

f n

Mx

Sl.1. Koncentisane sile na krutoj grediFig.1. Concentric forces on solid beam

Mjera varijabilnosti (rasipanja) slu%ajnih varijabli se defi-

niše tako'e iz statike, analogijom sa momentom inercije,

( )∑=

−=n

i

ii y f x x I

1

2 (12)

Na osnovu izraza (10) i (11) varijanca se možeizraziti kao o%ekivanje,

Analogy with the moment of inertia can be defined

measure of variability (dispersion) of random variables

[ ] ( )∑=

−= N

i

iii f x x xV

1

2 (13)

On the basis of expression (10) and (11) thevariance can be expressed as an expectation,

[ ] [ ] [ ]( )22 iii x E x E xV −= (14)

Standardna devijacija je: [ ] [ ]ii xV x =σ Drugi važan parametar rasipanja rezultata je

koeficijent varijacije, koji predstavlja mjeru odstupanja odcentralne osi,

Standard deviation is: [ ] [ ]ii xV x =σ Another important parameter of random variables

scattering is the coefficient of variation, which is ameasure of deviation from the central axis,

( ) ( )

( ) ( )%100⋅= x E x

xV σ

(15)

Za analizu kontinuirane raspodjele slu%ajno promjen-ljive može se primjeniti analogija sa jedankopodijeljenimvertikalnim optere(enjem na krutoj gredi od x(a) do x(b)[14], prikazano na slici 2.

For the analysis of continuous distribution of randomvariable can be applied analogy with distributed verticalload on solid beam, from x(a) to x(b) [14], presented inFig.2.

x

f(x)

x(b)x(a)

Mx

x1 x2

dx

Sl.2 - Jenoliko podijeljeno vertikalno optere' enje na krutoj grediFig.2 - Continuous distribution of vertical load on solid beam

U ovom slu%aju, izraz (9) je oblika, In this case, expression (9) has the form,

∫ =)(

)(

)(

b x

a x

dx x f M (16)


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a izraz (10), and expression (10),

M

dx x xf

x

b x

a x

∫ =

)(

)(

)(

(17)

Površina ispod krive f(x) za interval dx je vjero-vatno(a da (e vrijednost x biti u intervalu od x 1 do x 2 :

Area under the curve f(x) for interval dx is the proba-bility that the value of x will be in the interval from x 1 to x 2 :

[ ]∫ ≤≤=2

1

21)(

x

x

x x x P dx x f (18)

Površina ispod krive u intervalu x(a) do x(b) mora biti1 (jedan).

Area under the curve in the interval of x(a) to x(b)must be 1 (one).

∫ =)(

)(

1)(

b

a

x

x

dx x f (19)

Za kontinuiranu raspodjelu o%ekivana vrijednost je, For continuous distribution expected value is

[ ]

∫ =

)(

)(

)(

b x

a x

dx x xf x E (20)

a varijanca, and variance,

[ ] ( )∫ −=)(

)(

2)(

b x

a x

dx x f x x xV (21)

3 FUNKCIJA GUSTO%E VJEROVATNO%E

Teorija vjerovatno(e razmatra vjerovatno(u odre'e-ne pojavnosti doga'aja i kvantificira mjere nesigurnostislu%ajnih doga'aja. Metodologija, kojom se mogurazmotriti slu%ajnosti ili nesigurnosti u podacima ili mode-

lima, poznata je kao analiza nesigurnosti ili stohasti%kaanaliza. Na slici 3 pokazane su razne metode analize

nesigurnosti zasnovane na prikazu nesigurnosti.

3 PROBABILITY DENSITY FUNCTION

Probability theory treats the likelihood of a givenevent's occurrence and quantifies uncertain measures ofrandom events. Methodologies, which can consider therandomness or uncertainty in the data or model, are

known as uncertainty analysis or stochastic analysis.Figure 3 shows various methods of uncertainty analysisbased on the representation of uncertainties.

Sl.3 - Kategorije analize nepouzdanosti[7]Fig.3 - Uncertainty Analysis Categories[7]

Probabilisti%ki pristup zasnovan je na teoretskimosnovama funkcije gusto(e vjerovatno(e i uvo'enjukoriš'enja slu%ajno promjenljivih za prikaz nesigurnosti,dok deterministi%ki (ne-probabilisti%ki) pristup koristineprecizno znanje o stvarnim vrijednostima parametara.

Funkcija gusto(e vjerovatno(e reprezentira relativnuu%estalost odre'ene realizacije slu%ajne promjenljive.

The probabilistic approach is based on the theo-retical foundation of the probability density functioninformation and introduces the use of random variablesto represent uncertainty, while the non-probabilisticapproach manages imprecise knowledge about the truevalue of parameters.

The probability density function represents the


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Osnova statisti%ke obrade podataka jeste definisanjezakonitosti (funkcije) raspodjele slu%ajno promjenljivih.Podaci utvr 'eni analizom rezultata eksperimenata imajuodre'enu zakonitost raspodjele. U postupku analizeraspodjela utvr 'enih podataka se upore'uje sa poznatimteorijskim zakonitostima raspodjele. Izbor najpovoljnijezakonitosti raspodjele zapo%inje postavljanjem hipotezeo mogu(em teorijskom zakonu raspodjele. Ocjena

postavljene hipoteze se vrši preko statisti%kih testovazasnovanih na maksimalno dozvoljenom odstupanju

empirijskog i teorijskog zakona raspodjele. Aproksimaci- ja eksperimentima utvr 'ene raspodjele nekom poznatomteorijskom raspodjelom omogu(ava korištenje analizepoznate teorijske raspodjele, što zna%ajno pojednostav-ljuje postupak ocjene pouzdanosti sistema (kons-trukcije).

Nekoliko kontinuiranih raspodjela igra zna%ajnuulogu u gra'evinarstvu. Najvažnije su jednolika, ekspo-nencijalna, gama, beta, Weibull-ova, binomna, lognor-malna i normalna raspodjela. Najjednostavniji oblik konti-nuirane raspodjele je jednolika sa funkcijom gusto(evjerovatno(e, srednjom vrijednosti i varijancom,

b xaab x f ≤≤−= ,1

)( (22)

[ ]2

ab x E

+= (23)

Kod eksponencijalne raspodjele vjerovatno(a da sene(e desiti slu%ajno promjenljiva X u vremenskomintervalu t je,

relative frequency of certain realization for randomvariables. The basis of statistical data processing is todefine the probability density function. Data determinedduring analysis of the experiments results haveparticular laws of distribution. In the process of analysisthe distribution of determined data comparisons withknown theoretical laws of distribution. Selecting the bestlaws of distribution begins by setting the hypotheses

about the possible theoretical distribution law. Evaluationof set hypothesis is done by statistical tests based on themaximum permitted deviations of the empirical andtheoretical laws of distribution. Approximation of thedistribution determined by experiments with a knowntheoretical distribution analysis allows the use of well-known theoretical distribution, which greatly simplifiesthe process of assessing the reliability of the system(structure).

Several continuous distribution play useful roles incivil engineering. The more important ones are theUniform, Exponential, Gamma, Beta, Weibull, Binom,Lognormal and Normal distribution.

The simplest type of continuous distribution is theuniform with probability distribution function, mean and

variance

[ ] ( )

12

2ab

xV −= (24)

By exponential distribution the probability of nooccurrences of the random variable X during a timeinterval t is

( ) t e X P λ−== 0 (25)

a kumulativna funkcija raspodjele je, and cumulative distribution function is

( ) t T et F t T P λ−−==≤ 1)( (26)

Ova raspodjela je primjenljiva za elemente koji nemijenjaju svoje karakteristike u veoma dugom eksploa-tacionom vijeku. Pogodna je za betonske konstrukcije.

Sa fizikalnog stanovišta, inžinjeri i nau%nici suustanovili da empirijske raspodjele mnogih prirodnihprocesa i procesa u konstrukciji približno odgovarajugama raspodjeli. Standardna gama funkcija raspodjelevjerovatno(e može se napisati,

This distribution is applicable for elements that do notchange their characteristics in a very long period ofexploitation. It is suitable for concrete structures.

From the physical viewpoint, engineers andscientists have found that the empirical distributions ofmany natural and structural processes closely resemblethe gamma. The standard gamma probability distributionfunction is written as

>≤Γ =

−−

otherwise

r and t for r

et t f

t r

0

00)()(

1

(27)

)(r Γ je standardna gama funkcija definirana kao, )(r Γ is complete (standard) gamma function defined as

otherwise

r for dt et r t r

0

0,)(0

1

=

>=Γ ∫

∞−−

(28)

Parametar r je poznat kao parametar oblika.Beta raspodjela se koristi kod eksperimenata koji se

ponavljaju N puta sa odvojenim izlazima za svakieksperiment, gdje je x broj uspješnih eksperimenata.Modeli sa beta raspodjelom imaju posebnu ulogu kodmetoda odlu%ivanja. Raspodjela slu%ajno promjenljivekoja poprima vrijednosti u odre'enom intervalu. Betafunkcija raspodjele vjerovatno(e data je izrazom,

The parameter r is known as the shape parameter.The beta distribution is used in the experiments

repeated N times with independent output for eachexperiment, where x is the number of successfulexperiments. The beta distribution models plays aspecial role in decision methods. The distribution ofrandom variable that takes value in the interval. The betaprobability distribution function is given by


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Sl.4 - Standardna gamma funkcija gusto' e vjerovatno' e [16]Fig.4 - Standard gamma probability density functions [16]

( ) ( )

otherwise

x for x x B

x f

0

0,0,10,1,

1),(

11

=

>>

−

=

−

λλλλ

β ββ

(31)

Sl.6 - Weibull-ova funkcija gusto' e vjerovatno' e [16]Fig.6 - Weibull probability distribution functions [16]

Dok se duktilni otkaz armiranobetonskog konstruk-tivnog elementa dešava simultano duž površine otkaza,okarakterisan Gauss-ovom raspodjelom %vrsto(e kon-

While ductile failure of reinforced concrete structureelements occurs simultaneously along the failure surfaceand is characterized by Gaussian distribution of


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strukcije, bez uzimanja efekta veli%ine elementa, nakvazikrti otkaz zna%ajno uti%e veli%ina elementa i zavelike elemente se pojavljuju ekstremne statisti%ke vrije-dnosti po modelu najslabije karike, što vodi, prema [2],do Weibull-ve raspodjele %vrsto(e konstrukcije (uzpretpostavku da se otkaz dešava sa po%etkom mikroprslina).

Lognormalna raspodjela je vrlo dobar model za

prou%avanje otkaza usljed zamora i njegove zna

%ajneprimjene u podru% ju održavanja. Lognormalna funkcija

raspodjele vjerovatno(e data je izrazom,

structural strength with no size effect, quasibrittle failurespropagates, exhibits a strong size effect and, at largesizes, follows extreme value statistics of the weakest-linkchain model, which leads, according to [2], to Weibulldistribution of structural strength (provided that failureoccurs at macro-crack initiation).

Lognormal distribution is a very good model for studying the failure of which is the cause of fatigue and

has significant application in the field of maintenance.The Lognormal probability distribution function is givenby

( )

−−

⋅⋅=

2

2

2

ln

2

1)(

σ

λ

πσ

x

e x

x f (32)

Normalna funkcija raspodjele vjerovatno(e data jeizrazom,

The Normal probability distribution function is givenby

( )

−−

=2

2

2

2

1)(

σ

πσ

x x

e x f (33)

U radovima [7], [14], [16], [19], [20] i [22] opisane sudetaljnije funkcije raspodjele slu%ajno promjenljive, kojese koriste za analizu pouzdanosti tehni%kih sistema.

Za analizu pouzdanosti gra'evinskih konstrukcijaobi%no se koristi normalna raspodjela vjerovatno(e uobliku,

In the papers [7], [14], [16], [19], [20], and [22] aredescribed in some detail the functions of distribution ofrandom variable, which is used to analyze the reliabilityof technical systems.

For reliability analysis of civil structures commonlyused normal probability distribution in form

( )

2

2

1

2

1

−−

⋅= σµ

πσ

x

e x f (34)

f(x)

x

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

1 2 3 4 5 6 7-1-2-3-4x=0

f(x)

x=4

σ(x)=(2)/21/2

σ(x)=1

σ(x)=1/2

f(t) ϕ(z)

0

0

-µ σ

µ

t-µ σZ =

t

Sl.7 - Neki oblici krive normalne raspodjele

Fig.7 - Some forms of normal distribution curves

Sl.8 – Funkcija gusto' e vjerovatno' e za standardnu normalnuraspodjelu

Fig.8 – Probability density function for standardized normaldistribution

Kumulativna funkcija normalne raspodjele može seocijeniti samo numeri%kim metodama. U svrhu evaluacijeu praksi se koristi standardizovana kriva sa transforma-cijom promjenljive X u Z :

σ

µ−= t

z (35)

Cumulative distribution function of normal distributioncan only be evaluated by numerical methods. In practiceone uses the standardized curve for the purpose ofevaluation with the transformation of the variate X toas follows:

( ) ∞


9/24


4 MJERE POUZDANOSTI

Normativi po kojima se vrši analiza nosivosti gra'e-vinskih konstrukcija zasnovani su na deterministi%kim isemi-probabilisti%kim prora%unskim procedurama, sausvojenim faktorima sigurnosti. Odabrani faktori sigur-nosti za pojedine prora%unske situacije su rezultat pret-hodno provedene analize uticajnih parametara primje-nom teorije pouzdanosti.

4 MEASURES OF RELIABILITY

Standards in use to perform analysis of civilstructures capacity are based on deterministic and semi-probabilistic design procedures, with the adopted safetyfactors. Selected safety factors for the individual designsituations are the result of a previously performedanalysis of influential parameters using the theory ofreliability.

f(R)

f(S)

f(S)

f(R)

S,R

A2A1

f(S1)

S1 dS

Sl.9 – Definicija pouzdanostiFig.9 – Definiton of reliability

Vjerovatno(a da je uticaj S1 u intervalu ds jednaka jepovršini A1, , i može biti izražena kao,

Probability that the action S1 is in the interval ds,equal to the area A1, , can be expressed:

( ) 111122

AdsS f ds

S sds

S P ==

+≤≤− (37)

Vjerovatno(a da je otpornost ve(a od uticaja S1 jednaka je površini A2 :

Probability that the resistance is greater than theaction S1 is equal to the area A2 :

( ) ( )∫ ∞

==−1

21

s

AdR R f S R P (38)

Pouzdanost je proizvod vjerovatno(e (37) i (38), Reliability is the product of probability (37) and (38),

( ) ( )∫ ∞

=1

1

s

dR R f dS S f dR (39)

odnosno opšti izraz za pouzdanost je oblika: and the general expression for the reliability is:

( ) ( )∫ ∫ ∫ ∞

∞−

∞

== dS dR R f S f dR R

s

ili ( ) ( )∫ ∫ ∫ ∞

∞− ∞−

== dRdS S f R f dR R

R

(40)

Uvo'enjem smjene S R−=

ξ ( 0>

ξ ) i standardizi-

rane slu%ajno promjenljiveξσ

ξξ −= Z funkcija pouzda-

nosti ima oblik:

With the introduction of S R−=

ξ ( 0>

ξ ) and

standardized random variableξσ

ξξ −= Z reliability

function has the form:

∫ ∞

−

−⋅=

ξσ

ξπ

dZ e R Z 2

2

1

2

1 (41)


10/24


f(z)

zµzPf

Z0

Sl.10 – Standardizirana funkcija gusto' e vjerovatno' eFig.10 – Standardized probability density function

Neadekvatnost sistema da ispuni zahtjeve, mjerenakao pf , vezana je sa dijelom raspodjele gdje mjerasigurnosti Z poprima negativne vrijednosti, a odovara- ju(a pouzdanost je,

The inadequacy of the system to meet the demand,as measured by pf , is associated with that portion of thedistribution of the safety margin wherein Z takesnegative values, and corresponding reliability is

f p R −= 1 (42)

Važna mjera adekvatnosti inžinjerskog projektovanjae indeks pouzdanosti, definisan kao odnos izme'usrednje vrijednosti i standardne devijacije mjeresigurnosti sistema.

An important measure of the adequacy of anengineering design is the reliability index, defined as theratio between the mean and standard deviation of thesafety margin of the system.

z

z

σ

µβ = (43)

Opšti izraz za indeks pouzdanosti u odnosu na prvadva momenta kapaciteta i funkciju zahtjeva može senapisati kao,

A general expression for the reliability index in termof the first two moments of the capacity and the demandfunction can be written as

Sl.11 – Snaga aproksimacije vjerovatno' e otkaza kao funkcija indeksa pouzdanosti za normalnu raspodjelu S i R [16]Fig.11 – Power approximation of the probability of failure as a function of the reliability index for normally distributed S

and R [16]

222 S S R RS R

S R

σσσρσ

µµβ

+−

−= (44)

Ocjena pouzdanosti podrazumjeva usporedbu prora-%unatog indeksa pouzdanosti β sa vrijednosti indeksapouzdanosti koji se smatra adekvatnim na osnovu pret-hodnih iskustava na datom sistemu. U tom smislu, morase uspostaviti odnos izme'u kapaciteta sistema i zahtje-va (uticaji). Ukoliko su kapacitet i uticaji jednaki sistem jeu grani%nom stanju, koje se može pisati u obliku,

The assessment of reliability mean comparing thecalculated reliability index β with that found to beadequate from previous experience for the givensystem. For this purpose, one must establish arelationship between the capacity of the system and thedemand (load). If the capacity and load equal, this is alimit state, which can be written in the form


11/24


0),( =−= S RS R g (45)

Sl.12 – Stanje otkaza, stanje sigurnosti i grani % no stanje [20]Fig.12 – Failure state, safe state and limit state [20]

Op(enito su u gra'evinarstvu svi problemi funkcijaviše slu%ajno promjenljivih, gdje je važno da sve slu%ajnopromjenljive imaju odgovaraju(u pouzdanost. Pouzda-nost sistema slu%ajno promjenljivih, kod kojeg svakielement mora funkcionisati da bi sistem funkcionisaopredstavlja se kao serijski spoj uticajnih parametara [19],[22]:

Generally in the civil engineering all the problems arefunction of several random variable, where it is importantthat all random variables have adequate reliability.Reliability of random variable system, where eachelement must function that the system functioned,present as a serial combination of influential parameters[19], [22]:

( )n x x x P R ∪∪∪−= ....1 21 (46)i generalno je sistem me'usobno nezavisnih slu%ajnopromjenljivih, za koje vrijedi izraz,

and generally is a system of mutually independentrandom variable, for which valid expression:

( ) ( ) ( ) ( )∏=

=

==ni

i

in x P x P x P x P R

1

21 .... (47)

Za m nezavisnih slu%ajno promjenljivih x i , sanormalnom raspodjelom, dobije se m-dimenzionalnaraspodjela gusto(e prema multiplikacionom praviluprora%una vjerovatno(e [12]:

For m independent random variable x i , with normaldistribution, the result is m-dimensional densitydistribution according to multiplication rule of probabilitycalculation [12]:

)2

1exp(

)2(

1)(..)()(),...,,(

1

2

1

2

221121 ∑=

=

−⋅−⋅

Π⋅

=⋅⋅⋅=m

i x

xi

x

m

i

mm xm x xm x

i

i

i

m x x f x f x f x x x f

σσπ

(48)

U slu%aju dvije osnovne promjenljive: x 1 i x 2 , odnosnoR (otpornost) and S (dejstva), dobiju se hiper površine,istih raspodjela gusto(e, odnosno m-dimenzionalnekugle [12] (Slika 10).

In the case of two basic variables: x 1 and x 2 , and R (resistance) and S (action), get a hyper surface, thesame distribution density, and m-dimensional balls [12](Fig.10).

Sl.13 - Jedna% ina grani % nog stanja u x domenu [12]Fig.13 - Limit state equation in x – area [12]


12/24


U praksi funkcija gusto(e nije poznata i najviše što jepoznato je set n slu%ajno promjenljivih X koje definirajuproblem konstrukcije (kao što su mehani%ke igeometri ske karakteristike, pokretna i stalna dejstva,dejstvo prednaprezanja itd.). taviše, kod prora%unabetonskih konstrukcija grani%na stanja su obi%noformulisana preko funkcija slu%ajno promjenljivih Y=Y(X)

koje opisuju odgovor konstrukcije (odnosno naprezanja,dilatacije itd.), i ovakvi izvodi su generalno dostupnisamo u implicitnom obliku. Stoga je neophodannumeri%ki pristup. Za rješavanje sistema slu%ajnopromjenljivih razvijene su 3 metode rješavanja:

1. Ta'na metoda, koja zahtjeva kao ulazne podatkepoznate funkcije distribucije vjerovatno(e za svepromjenjive. U postupku rješavanja primjenjuje senumeri%ka integracija i Monte Carlo metoda. Ovajpostupak zahtjeva opsežnu kompjutersku analizu.

2. FOSM (First order second moment method), gdjese u postupku rješavanja koriste Taylor-ovi redovi.

3. PEM (Point estimate method) koja se najvišekoristi u analizama jer omogu(ava postepeni razvoj

rješenja. Analiti%ka integracija je mogu(a samo kod specijalnihslu%ajeva. Numeri%ko rješenje problema je jednostavnije,ali ako je pod integralom više od dvije slu%ajnopromjenljive, numeri%ka integracija ne može u svimslu%ajevima dati dovoljno ta%no rješenje. Sa pove(anjembroja varijabli zna%ajno rastu prora%unski zahtjevi, jer suu tim slu%ajevima integraciona podru% ja složenijegeometrijske figure, n-dimenzionalne sfere. Stoga su zaodre'ene metode razvijena rješenja na osnovunumeri%ke integracije, kao što su Monte Carlo metode,koje mogu biti primjenjene samo kod razvijenihsistemati%nih metoda numeri%kog uzorkovanja osnovnihpromjenljivih X, kao što je Markov lanac metodasimulacije prezentirana u radu [3].

U radu [1] opisane su dvije metode Monte Carlosimulacije: direktna metoda simulacije i uzorkovanjeprema važnosti. Direktna metoda simulacije obuhvatauzorkovanje osnovnih nekoreliranih varijabli premanjihovim odgovaraju(im probabilisti%kim karakteristikamai njihovo uvo'enje u funkciju performansi Z , koja možebiti izražena u funkciji od osnovnih slu%ajno promjenljivih X i za odgovaraju(a dejstva i %vrsto(u konstrukcije

In practice the density function is not known and atthe most some information is available only about a setof n basic random variables X which define the structuralproblem (e.g. mechanical and geometrical properties,dead and live loads, prestressing actions, etc.).Moreover, in concrete design the limit states are usuallyformulated in terms of functions of random variables

Y=Y(X) which describe the structural response (e.g.stresses, strains, etc.), and such derivation is generallyonly available in an implicit form. Therefore a numericalapproach is required. For solving random variablesystems have been developed 3 methods of solution:

1. Accurate method, which requires as input knownprobability distribution functions for the variable. In theprocess of resolving apply numerical integration andMonte Carlo methods. This procedure requires extensivecomputer analysis.

2. FOSM (First order second moment method),where in the process of solving the use Taylor's series.

3. PEM (Point estimate method) that is most used inthe analysis because it allows a gradual development of

solutions. Analytical integration is possible only in specialcases. Numerical solution of the problem is simpler, butwhen it is integral to more than two random variablesnumerical integration can not in all cases providesufficient accurate solution. With the increasing numberof variables significantly increasing the calculationrequirements, because in these cases integration areasare complex geometric figure, n-dimensional sphere.Therefore, solutions based on numerical integration forthe specific methods were developed, such as MonteCarlo simulation, which can be applied only to developedsystematic methods of numerical sampling of basicvariables X, or Markov chain simulation methodpresented in the paper [3].

The paper [1] described two Monte Carlo SimulationMethods: the direct simulation method and importancesampling. The direct simulation method comprisesdrawing samples of the basic noncorelated variablesaccording to their corresponding probabilisticcharacteristics and then feeding them into theperformance function Z , that can be expressed in termsof basic random variables X i for relevant loads andstructural strength

S R X X X X Z Z n −== ),...,,,( 321 (49)

Srednja vrijednost i varijanca vjerovatno(e nezado-voljavaju(ih performansi,

The mean and the variance of the unsatisfactoryperformance probability

∫ ∫ = nn X X u dxdx x x f P n ...),...,(... 11...1 (50)

može se izraziti kao

N

N P uu = (51)

uz pretpostavku da je N u broj ciklusa simulacije za Z


13/24


njim vrijednostima koje su bliže prora%unskim vrijedno-stima nego njenoj originalnoj raspodjeli vjerovatno(e.Jedna%ina (50) za ovu metodu ima oblik

mean values that are closer to the design value thantheir original probability distributions. Equation (50) forthis method have form

∑=

= N

i nii X

nii X iu

x xh

x x f I

N P

1 1

1

),...,(

),...,(1 (53)

gdje jeI – funkcija indikatora performansi sa vrijednostima 0

ili 1f X – originalna funkcija gusto(e osnovnih slu%ajno

promjenljivihh X – odabrana funkcija gusto(e osnovnih slu%ajno

promjenljivihPrimjena ove metode prezentirana je u radu [11].Primjena Monte Carlo simulacije za analizu

pouzdanosti armiranog betona i prednapregnutogbetona data je u radu [4], na primjeru lu%nog mosta.

U radu [5] opisana je primjena Fuzzy teorije uodnosu na teoriju vjerovatno(e. Fuzzy teorijaomogu(ava analizu nesigurnosti u slu%aju nedostatkainformacija za razliku od teorije vjerovatno(e, koja jezasnovana na potpunom znanju o stohasti%kim

promjenama koje su rezultat slu%ajne prirode odre'enihveli%ina. Korištenjem Fuzzy kriterija sve nesigurnostimogu se modelirati kroz interval vrijednosti, ograni%enodgovaraju(im minimalnim i maksimalnim ekstremom.Fuzzy funcija pripadnosti )(~ x

Aµµ = fuzzy seta

X A ⊆~

, za odre'eni nivo pripadnosti [ ]1,0∈α , jemogu(a raspodjela pogodna za opis informacija onesigurnosti, tamo gdje raspodjela vjerovatno(e nijedirektno dostupna.

whereI – performance indicator function that takes value of

0 or 1f X – the original density function of the basic random

variablesh X – the selected density function of the basic

random variables Application of these methods is presented in the

paper [11]. Application of Monte Carlo simulation for reliability

analysis of reinforced concrete and prestressed concreteis provided in the paper [4], for example arc bridge.

The paper [5] describes the application of fuzzytheory in relation to the probability theory. Fuzzy theoryallows analysis of uncertainty in case of lack ofinformation as opposed to the probability theory, which is

based on a perfect knowledge about the stochasticvariability resulting from the random nature of the samequantities. Using a fuzzy criterion all uncertainties can bemodeled through bands of values, bounded betweensuitable minimum and maximum extremes. A fuzzy

membership function )(~ x A

µµ = of a fuzzy set X A ⊆~

,

for defined level of membership [ ]1,0∈α , is apossibilistic distribution suitable to describe uncertaininformation, when a probabilistic distribution is notdirectly available.

Sl.14. (a)Pripadaju' a funkcija i α nivo; (b)Mapiranje izme4 u intervala nepouzdanosti na osi x i odgovaraju' eg intervalaodgovora na osi λF [5]

Fig.14. (a)Membership function and α level; (b)Mapping between the interval of uncertainty on x and the correspondingresponse interval on λF [5]

Primjena Taylor-ovih redova zahtjeva formulisanje irješavanje derivacija, što je za viševarijabilne problemezahtjevan zadatak, posebno kada je funkcija zadanaimplicitno u vidu krive, grafa ili kao rješenje MKE.Pojednostavljeni model pouzdanosti, prvobitno pred-stavljen kod projektovanja %eli%nih konstrukcija, koristisamo srednju vrijednost i koeficijent varijacije zaotpornost R i dejstva S u pojedinim grani%nim stanjimada bi se dobio indeks pouzdanosti β, koji se prora-%unavao kao,

Application of Taylor's series requests in formulatingand solving the derivative, which is for multi-variableproblems demanding task, especially when the functionis implicit in the form of the default curve, a graph or as aFEM solution. A simplified reliability model, firstintroduced in structural steel design, only uses the meanvalues and coefficients of variation for the resistance Rand load S in a particular limiting state to obtain thereliability index β, which is computed as


14/24


22

)/ln(

S R

S R

V V +=

µµβ (54)

nezavisno od tipa raspodjele R i S.Primjeri su prezentirani u radovima [7], [14] i [16].PEM metodu je prezentirao Rosenblueth 1975 g., a

potpuno je razvijena 1981 g. Metoda je zasnovana naanalogiji izme'u distribucije vjerovatno(e i distribuiranog

vertikalnog optere(enja na horizontalnoj krutoj gredi.O%ekivana vrijednost odgovara položaju djelovanja silekoja uravnotežuje optere(enje (jedini%na vrijednost) ilitežištu optere(enja. Rosenblueth je predložio da separametri o%ekivane vrijednosti i standardne devijacijeodrede iz analogije sa slobodno oslonjenom gredom.

regardless of the type of distribution of R and S.Examples are presented in the papers [7], [14] and

[16].PEM method was presented by Rosenblueth 1975,

and is fully developed in 1981. Method is based on the

analogy between probability distributions and distributedvertical load on the horizontal solid beam. Expectedvalue corresponds to the position of action that balancesthe load force (unit value) or the center of gravity of theload. Rosenblueth proposed that the parameters of theexpected value and standard deviation determineaccording to the analogy with the simple beam.

f(x)

x

a b

E[x]

x

p

x

p

Sl.15 – Distribucija vjerovatno' e – analaogija sa krutom gredomFig.15 – Distribution of probability – solid beam analogy

Reakcije p- i p+ su dvije ravnotežne ta%ke funkcijedistribucije f(x). Primjenom postavki teorije pouzdanostimogu se postaviti slijede(e jedna%ine:

• Uslov ravnoteže:

Reactions p- and p + are two points of equilibrium ofdistribution function f(x). Applying the theory of reliabilitycan set the following equation:

• The condition of equilibrium:1=+ −+ p p (55)

• O%ekivana vrijednost: • Expected value:

( )[ ] x x f E x p x p ==+ −−++ (56)

• Mjera rasipanja • The measure of dispersion

( ) ( ) ( )[ ] [ ] x x f x x p x x p 2222 σσ ==−+− −−++ (57)

• Asimetrija raspodjele: • Asymmetry of distribution:

( ) ( ) ( ) [ ] x x x p x x p 333 1σβ=−+− −−++ (58)

Rješenje jedna%ina (55) do (58) su: Solutions of equations (55) to (58) are:

( )

+

−±=+ 2

2

11

111

2

1

β p ; +− −= p p 1 ; [ ]

+

−+ +=

p

p x x x σ ; [ ]

−

+− −=

p

p x x x σ (59)

Kod standardizirane normalne raspodjele može seusvojiti ( ) 01 =β tako da izrazi (59) imaju oblik:

With standardized normal distribution can beadopted ( ) 01 =β so that expressions (59) have theform:

2

1== −+ p p ; [ ] x x x σ+=+ ; [ ] x x x σ−=− (60)


15/24


Na osnovu odre'enih ta%aka slu%ajno promjenjivih x ifunkcionalne zavisnosti x i y dobiju se vrijednosti funkcijey(x), y + i y-, korištenjem izraza,

On the basis of certain points of random variable x and the functional dependence x and y get the value ofthe function y(x), y + and y-, using the expression,

M M M y p y p y E ++−− += (61)

gdje M odgovara broju poznatih momenata za slu%ajnopromjenjivu x .

5 ANALIZA POUZDANOSTI VEZEPREFABRIKOVANE PLO"E I MONOLITNOG ZIDA

U periodu 2004. – 2008. godina provedena su eks-perimentalna i numeri%ka istraživanja veze prefabrikova-ne plo%e i monolitnog zida (slika 16), sa ciljem defini-sanja mehanizma rada ovakve veze. Detalji istraživanja irezultati istraživanja prezentirani su u radovima [23] i[24].

Numeri%ki modeli su ura'eni korištenjem MKE i Linkelemenata modeliranih korištenjem eksperimentomdobijenog M-φ radnog dijagrama (slika 17) [24].Uporedni rezultati provedenih numeri%kih istraživanjaprezentirani su na slici 18. Istraživanjem je utvr 'eno da i

monolitne veze imaju odre'en stepen popustljivosti. Izmnoštva rezultata ovdje se daju dvije vrijednosti stepenapopustljivosti prefabrikovane veze u odnosu namonolitnu:

• Eksploataciono optere(enje - γ calc.,serv. = 0.92• Grani%no optere(enje sa faktorom sigurnosti 1.75 -

γ calc.,1,75 = 0.90

where M corresponds to the number of known momentsfor random variable x .

5 RELIABILITY OF THE PRECAST PLATE-MONOLITHIC WALL CONNECTION

In the period 2004th-2008th years, were performedexperimental and numerical research of precast plate –monolithic wall connection (fig.16), with the aim ofdefining the work mechanism of such connections.Details of the research and results are presented in thepapers [23] and [24].

Numerical models are done using FEM and Linkelements modeled in the experiment obtained using M-φworking diagram (Fig.17) [24]. Comparative results ofthe conducted numerical studies are presented in Figure18. Research has found that the monolithic connection

have a certain degree of yielding. From the many resultshere are two values of the degree of prefabricatedconnections yielding relatively to the monolithic:

• Serviceability load - γ calc.,serv. = 0.92• Ultimate load with safety factor 1.75 - γ calc.,1,75 =

0.90

Pw=15.75kN(one floor)...78.75kN(five floors)

M o n o l i t h i c w a l l

Model width B=50cm

Pre-cast plate

Pk

M o n o l i t h i c w a l l

Sl. 16 – Montažna veza montažne plo% e i monolitnog zidaFig. 16 – Precast connections of precast plate and monolithic wall

U radu [23] detaljno su dati rezultati eksperimen-talnog istraživanja. Ovdje se daju uporedne vrijednostistepena popustljivosti prefabrikovane veze u odnosu namonolitne:

• γ exp,serv. = 0.907 ÷ 0.913• γ exp,1,75 = 0.897 ÷ 0.903

The paper [23] have detailed the results of experi-mental research. Here give comparative values of thedegree of prefabricated connections yielding relatively tothe monolithic:

• γ exp,serv. = 0.907 ÷ 0.913• γ exp,1,75 = 0.897 ÷ 0.903


16/24


8 m

q h z

= 2 ,

Sl. 17 – MKE model [24] Fig. 17 – FEM Model [24]

ABSOLUTELY RIGIDMONOLITHIC YIELDING

5

10

15

20

25

30

35

40

45

50

55

1 2 3 4 5

M (kNm)

5

10

15

20

25

30

35

40

1 2 3 4 5 1514131211106 7 8 9 p (kN/m)1614131211106 7 8

PRECAST

MATRIX K AND "BEAM"ELEMENTS MODEL "SHELL" ELEMENTS MODEL

ABSOLUTELY RIGIDMONOLITHIC YIELDINGPRECAST

Sl.18. Ra% unske krive oslona% ki moment - optere' enje (M-p) za analizirane numeri % ke modeleFig.18. Calculation curves support moment – load (M-p) for analyzed numerical models

Uporedba rezultata pokazuje zadovoljavaju(u ta%nostnumeri%kih modela. Da bi se verificirali prezentiranirezultati neophodna je analiza pouzdanosti dobijenih

karakteristi%nih vrijednosti u cilju definisanja stepenasigurnosti (faktora sigurnosti) analiziranog prefabrikova-nog sistema gra'enja.

5.1 Analiza popustljivosti veze

Provedena analiza pouzdanosti stepena popustlji-vosti istraživane veze prefabrikovane plo%e i monolitnogzida definisana je izrazom datim u radu [8],

Comparison of results showed a satisfactory ac-curacy of the numerical model. To verify the presentedresults is necessary reliability analysis of the obtained

characteristic values with the aim of defining the safetydegree (factor) of analyzed prefabricated buildingsystems.

5.1 Analysis of the connection yielding

Reliability analysis of yielding degree of researchedprecast plate-monolithic wall connection, defined byexpression given in the paper [8], was performed

.

.

.

.

pl

pl

pl

pl

I E

LS

B

LS

⋅⋅

=⋅

= φφα (62)


17/24


Analiza je ura'ena uz pretpostavku normalnefunkcije raspodjele vjerovatno(e za uticajne parametre.Raspon plo%e je usvojen konstantan L pl =6,15m, a koe-ficijent iskrivljenosti raspodjele β1=0 . Dakle, analiziran jestepen popustljivosti kao funkcija tri slu%ajno pro-mjenljive I E S f ,,φα = .

Pošto e ispitivanje provedeno na 3 monolitna i 3montažna modela ne postoji dovoljan broj podataka za

statisti%ku obradu podataka, odnosno pouzdanu deter-minaciju parametara rasipanja rezultata. Stoga suobra'eni rezultati ispitivanja upore'ivani sa do sadaprovedenim istraživanjima i preporukama prezentiranimu [6], [9], [14], [18] i [19].

Tako je Ellingwood 1980.godine, na osnovu prove-denih opsežnih eksperimenata, preporu%io reprezen-tativne vrijednosti koeficijenta varijacije, i to za betonskeelemente izložene savijanju 8÷9,5%. Isti autor je defi-nisao za spoj greda-stub koeficijent varijacije 10%.Mehlhorn je u [18] prezentirao opsežna istraživanjasmicanja glatkih spojnica gdje je za 176 opita dobijenkoeficijent varijacije 16%. Iste rezultate su dobili Hansen,Olesen, Fauchart i Cortini. Pommeret je dobio koeficijent

varijacije 12÷15%, Laing 17÷18%, Pume 13%. U radu[13] je prezentirana statisti%ka obrada rezultata ispitanih95 modela za koje je dobijen koeficijent varijacije 12%.

Na osnovu prethodnih navedenih podataka i rezulta-ta eksperimenata analiza je provedena za koeficijentevarijacije 10 - 25 %. Za o%ekivanu vrijednost je usvojenaeksperimentima dobijena srednja vrijednost

rad kNmS /94,3507=φ .Kod proizvodnje prefabrikovanih elemenata veli%ina

dopuštene greške zavisi od tipa konstrukcije, na%inaproizvodnje i montaže elemenata i naj%eš(e se kre(e ugranicama od 3 ÷10 mm Error! Reference source notfound..

Za potrebe definisanja pouzdanosti u ovom radu

usvojena je tolerancija za širinu plo%e modela ±10mm, tj.mmb pl 10500. ±= , a za debljinu plo%e ±10mm, tj.

mmd pl 3160. ±= .Uz pretpostavku standardizovane normalne raspo-

djele i karakteristi%nih 5%-fraktilnih vrijednosti dozvolje-nih odstupanja, koeficijenti varijacije su:

− za širinu modela CV = 1,22 % (usvojeno 1,5%)− za debljinu plo%e CV = 1,13 % (usvojeno 1,5%)Razlika eksperimentom utvr 'enih reprezentativnih

vrijednosti modula elasti%nosti betona monolitnih imontažnih modela je 4,78%. Ova razlika ima odraz narazli%ite deformacione karakteristike. Me'utim provede-na eksperimentalna i numeri%ka istraživanja su pokazalada utvr 'ena veli%ina razlike nije imala zna%ajan uticaj nauporednu analizu stepena popustljivosti monolitnih mon-tažnih modela. Pošto su izrazi za odre'ivanje modulaelasti%nosti u funkciji %vrsto(e na pritisak betona, ovdjese u nedostatku dovoljnog broja podataka uzimajupreporu%ene vrijednosti koeficijenata varijacije, zasno-vane na opsežnim istraživanjima pojedinih autora.

Ellingwood je 1980. god. predložio za armiranibetonski element optere(en na savijanje koeficijentvarijacije 14% [14]. Bljuger je prilikom analize spojevazidova usvojio koeficijent varijacije CV=12,8% [6].Zasnovano na rezultatima istraživanja koja su uradiliEntroy 1960, Murdock 1953, Rusch 1969, Mirza 1979,

The analysis is done assuming a normal probabilitydensity function for the influential parameters. The spanof plate was adopted L pl =6,15m, and coefficient ofdistribution distortion β1=0. Therefore, the yieldingdegree as a function of three random variables

I E S f ,,φα = was analyzed.Because the testing performed on the 3 monolithic

and 3 precast models 3 was not enough data for

statistical data processing, and reliable determination ofscattering results parameters. Therefore, the obtainedresults of experimental research compared with resultsof so far performed research and recommendationspresented in [6], [9], [14], [18] and [19].

On the basis of extensive experiments Ellingwood1980’s recommended a representative coefficient ofvariations, for concrete elements exposed to bending,from 8 to 9.5%. The same author has defined for thebeam-column connection coefficient of variation of 10%.Mehlhorn in [18] presented extensive studies of smoothshear connectors, where for the 176 experimentsobtained coefficient of variation 16%. The same resultswere obtained Hansen, Olesen, Fauchart and Cortini.

Pommeret got a coefficient of variation 12-15%, Laing 17÷18%, Pume 13%. The paper [13] presented a statisticalanalysis of the testing results with 95 models, for whichthe obtained coefficient of variation of 12%.

Based on these previous data and results ofexperiments, analysis for the coefficient of variation from10 to 25% was performed. The experiment obtainedmean value rad kNmS /94,3507=φ was adopted as the

expected value.By production of precast elements, size of permitted

deviation depends on the type of structures, productionprocedure and erection procedure, and often is withinthe limits of 3 to 10mm Error! Reference source notfound..

For the purposes of the definition of reliability in thispaper, tolerance for the width of panel models wasadopted ±10mm, ie., mmb pl 10500. ±= , and for a plate

thickness ±10mm, ie., mmd pl 3160. ±= .

Assuming standardized normal distribution andtypical 5%-fractil values of permitted deviations,coefficients of variation are:

− for width of the model CV = 1.22 % (adopted 1.5%)− for plate thickness CV = 1.13% (adopted 1.5%)Difference of value of modulus of elasticity of precast

and monolithic model concrete determined with expe-riment is 4.78%. This difference is reflected in differentdeformation characteristics. However, conducted expe-

rimental and numerical studies have shown that definedsize of differences had no significant effect on the com-parative analysis of the yielding degree of monolithic andprecast models. Since the expressions for determiningthe modulus of elasticity in the function of compressivestrength of concrete, here is the lack of sufficient data totake the recommended values of coefficients of varia-tions, based on extensive research of individual authors.

Ellingwood in 1980. proposed for reinforced concreteelements loaded in bending coefficient of variation of14% [14]. Bljuger during the analysis of compound wallsadopted coefficient of variation CV = 12.8% [6]. Basedon the results of research that is done Entroy 1960,


18/24


Melchers je prezentirao u tabeli 8.7 [19] koeficijentevarijacije, odnosno standardne devijacije koje se kre(uod 2,8 MPa za odli%an beton do 5,6 MPa za loš beton.Ovo su rezultati za %vrsto(e na pritisak fc > 28 MPa. Za%vrsto(u na pritisak nominalne vrijednosti 30 MPakoeficijent varijacije je: CV=9,33% za odli%an beton, aCV =18,66% za loš beton.

Za analizu pouzdanosti u ovom radu usvojeni su

koeficijenti varijacije kao i kod krutosti spoja 10 - 25 %.Usvojena o%ekivana vrijednost je 2/34975 m MN E c = .Za funkciju tri slu%ajno promjenljive y = y (x 1, x 2 , x 3 )

vrijedi:

Murdock 1953, Rusch 1969, Mirza 1979, Melchers ispresented in Table 8.7 [19] coefficient of variation orstandard deviation ranging from 2.8 MPa for the excel-lent concrete to 5.6 MPa for the bad concrete. These arethe results of compressive strength fc > 28 MPa. Forcompressive strength with a nominal value 30 MPa thecoefficient of variation is CV = 9.33% for the excellentconcrete, and CV = 18.66% for the bad concrete.

For reliability analysis in this paper have beenadopted the coefficient of variation as for connectionstiffness from 10 to 25%. Expected value is adoptedexperimental mean value 2/34975 m MN E c = .

For the function of three random variable y = y (x 1, x 2 , x 3 ) is valid:

])[],[],[( 332211 x x x x x x y y σσσ ±±±=±±± (63)

Težinski koeficijenti p su: Weight coefficients p are:

( )3123123 12

1ρρρ +++== −−−+++ p p ; ( )3123123 12

1ρρρ −−+== +−−−++ p p

( )3123123 1

2

1ρρρ +−−== −+−+−+ p p ; ( )3123123 1

2

1ρρρ −+−== ++−−−+ p p (64)

O%ekivana vrijednost je: Expected value is: M M M M y p y p y p y E −−−−−−−++−++++++++ +++= ...][ (65)

Pošto je moment inercije I funkcija dvije slu%ajnoprojenljive I = f(b pl , h pl ), prethodno je ura'ena analizastatisti%kih parametara momenta inercije. Statisti%komobradom podataka za širinu modela i debljinu plo%edobijen je koeficijent korelacije: 0.. ≈hpl bpl ρ . Prema tome

težinski koeficijenti su:

Since the moment of inertia I is function of tworandom variable I = f(b pl , h pl ), analysis of the preliminarystatistical parameters of the moment of inertia wereperformed. Correlation coefficient of model width andthickness of plate was adopted 0.. ≈hpl bpl ρ . Accordingly,

weight coefficients are:

4

1==== −−+−−+++ p p p p ;

12

3. pl pl hb

I ⋅

=

a prora%unata varijacija, and calculated variation is( ) %97,5= I V

U daljnjem prora%unu usvojena je varijacija V(I) =6,0%.

U radu je prezentirana analiza jednog statisti%kogslu%aja. Pregled svih analiziranih statisti%kih slu%ajevadaje se u tabeli 1.

Slu'aj (a)

In further calculations was adopted V(I) = 6,0%.The paper presented the statistical analysis of one

case. Review of the all statistical cases is given inTable 1.

Case (a)

400003392,0 m I = ( ) %6= I V

2/34975 m MN E c = ( ) %10=c E V

rad kNmS /94,3507=φ %6=φS V L=6,15 m=const.

Koeficijenti korelacije eksperimentalnih rezultata su:6,0;9,0;4,0 ,,, === φφ ρρρ S E S I E I C

Correlation coefficients of experimental results are:6,0;9,0;4,0 ,,, === φφ ρρρ S E S I E I C

73,38585,38472000035955,0 === +++ φS E I C

15,31575,31477000031884,0 === −−− φS E I C

Težinski koeficijenti su: Weighting factors are:

( ) 3625,06,09,04,018

1=+++== −−−+++ p p


19/24


( ) 0125,09,06,04,018

1−=−−+== +−−−++ p p

( ) 1125,09,06,04,018

1=+−−== −+−+−+ p p

( ) 0375,09,06,04,018

1=−+−== ++−−−+ p p

( )

183,334227,18

6721,135268,37401293,73625,0346,19

9886,6086,55929556,00125,0645,23

1877,28557,25078076,11125,0829,15

0351,14268,374725475,00375,0346,19

0373,11328,29464335,00375,0156,17

4614,49657,4393589,21125,0968,20

4287,2296,19417424,00125,0939,13

2066,105225,29017555,63625,0036,17,,

22

−−−

+−−

−+−

++−

−−+

+−+

−++

+++

−−−

−−−

⋅⋅

α

α

α

α

α

α

α

αααααα φ ijk ijk ijk ijk ijk ijk ijk C p p pS E I

[ ] [ ] 183,334;227,18 2

== αα E E [ ] ( ) 95947,1227,18183,334 2 =−=αV [ ] 3998,1=ασ

Varijacija stepena popustljivosti je: Variation of yielding degree is:

( ) %68,7100227,18

3998,1=⋅=αV

Na osnovu provedene statisti%ke analize odre'ena jeminimalna vrijednost stepena popustljivosti za koju jevjerovatno(a manjih vrijednosti 5 % (karakteristi%na 5%-fraktilna vrijednost). Prora%un je proveden primjenomstandardizovane normalne raspodjele.

On the basis of the statistical analysis determinedthe minimum value of the degree of yielding, for which isprobability of smaller value of 5% (typical 5% fractilvalue). The calculation is performed using thestandardized normal distribution.

931,153998,164,1227,1805,0 =⋅−=α Za prora%unatu vrijednost oslona%ki moment (e biti: For above calculated value moment on support will

be:

M M osl ⋅= 888,0.

Tabela 1 – Analizirani statisti % ki slu% ajeviTable 1 – Analyzed statistical cases

Moment of inertia I (m4)Modulus ofelasticity Ec

(MN/m2)

Connection stiffnessSφ (kNm/rad)

Degree ofyielding γ 1 Statictical

cases

E[I] V(I)% E[Ec] V(Ec)% E[Sφ] V(Sφ)% V(γ )% γ 0,05

Case (a) 0,00003392 6,0 34975 10,0 3507,94 10,0 7,68 0,888Case(b)

0,00003392 6,0 34975 15,0 3507,94 15,0 11,31 0,878

Case(c)

0,00003392 6,0 34975 20,0 3507,94 25,0 26,17 0,839

Case(d)

0,00003392 6,0 34975 25,0 3507,94 25,0 20,61 0,860

Case(e)

0,00003392 6,0 34975 10,0 3507,94 25,0 15,90 0,869

Case (f) 0,00003392 6,0 34975 25,0 3507,94 10,0 21,91 0,860

1 α

αγ

+=

2


20/24


Na osnovu provedene analize usvojena je maksi-malna o%ekivana varijacija 27 %, odnosno zaokruženanajve(a vrijednost iz tabele 1.

Karakteristi%na 5 % fraktilna vrijednost α je,

Based on the performed analysis adopted themaximum expected variation of 27%, ie., maximumvalue from Table 1.

Characteristics 5%-fractil value of α is,

183,102935,464,1276,1805,0 =⋅−=α

odnosno moment nad osloncem, or the moment on support is,

M M E osl ⋅= 835,0)( 05.0., Varijacija momenta nad osloncem je:

%22,7)( 05.0., =osl M V .

5.2 Analiza vjerovatno)e otkaza veze

Ellingwood je 1980. god., na osnovu statisti%ke obra-de rezultata mjerenja na objektima, dao prijedlog repre-zentativnih vrijednosti varijacija za stalno i pokretnooptere(enje i to:

− za stalno optere(enje V(g) = 10%− za pokretno optere(enje V(p) = 25%Pošto je kod optere(enja ispitanih spojeva odnos g:p

= 2:1, varijacija ukupnog optere(enja je V(q) = 15%.O%ekivana vrijednost momenta nad osloncem (u spoju),za eksploataciono optere(enje, je E(M q)=14,17 kNm.

Uz usvojene varijacije za stalno i pokretnooptere(enje, grani%ne vrijednosti uticajnih momenata suM q,min=12,04 kNm i M q,max =16,30 kNm.

Srednja vrijednost momenta nosivosti utvr 'ena

eksperimentom je: kNm M n 67,42= Uz uvažavanje analize pouzdanosti stepena

popustljivosti spoja, provedene u prethodnoj ta%ki,minimalna i maksimalna vrijednost momenta nosivostispoja je M n,min=39,58 kNm i M n,max =45,75 kNm.

Mjerodavni statisti%ki parametri su:

kNm M M qn z 5,2817,1467,42 =−=−=µ 222 Mq Mn z σσσ +=

( ) ( )222 491,967,420722,0 kNm Mn =⋅=σ

Indeks pouzdanosti je,

Variation of moment on support is:%22,7)( 05.0., =osl M V .

5.2 Analysis of the probability of connection failure

On the basis of statistical processing results ofmeasurements on objects, Ellingwood 1980th proposedrepresentative values of variation for dead and live loadsas follows:

− for dead load V(g) = 10 %− for live load V(p) = 25 %Since the load ratio of researched connection g:p =

2:1, the variation of the total load is V(q) = 15%.Expected value of moment on support (in connectionarea), for service load, is E(M q )=14,17 kNm.

With variations adopted for the dead and live loads,the limit value of moments are M q,min=12,04 kNm andM q,max =16,30 kNm.

Average value of moment capacity determined in the

experiment is: kNm M n 67,42= With the appreciation of the reliability analysis of

degree of yielding, implemented in the preceding para-graph, the minimum and maximum value of connectionmoment capacity is M n,min=39,58 kNm and M n,max =45,75kNm.

The relevant statistical parameters are:

( ) ( )222 518,417,1415,0 kNm Mq =⋅=σ

009,14518,4491,92 =+= z σ

Reliability index is,

61,7743,3

5,28==β

odnosno vjerovatno(a otkaza je, or probability of failure is,

( ) 1410445,161,7 −⋅=−= φ f P .

Eksperimentom je utvr 'ena varijacija momenta loma10 %. U tom slu%aju je indeks pouzdanosti,

Experimental value of variaton of failure moment is10%. In this case, the reliability index is,

98,5767,4

5,28 ==β

a vjerovatno(a otkaza, and the probability of failure,

( ) 910325,198,5 −⋅=−= φ f P .

U tabeli 2 date su preporu%ene minimalne vrijednostiindeksa pouzdanosti za klase pouzdanosti, preuzete izreference[10].

Indeks pouzdanosti je definisan za po%etnu krutostprefabrikovane veze. Ako je indeks pouzdanosti β < 2, uradu [15] preporu%uje se uvo'enje u analizu rezidualnekrutosti veze.

In table 2 are given recommended minimum valuesof reliability index for reliability classes RC, taken fromreference [10].

Reliability index is determined by the initial stiffnessof the precast connection. If the reliability index β < 2,the paper [15] recommends the introduction to theanalysis of residual connection stiffness.


21/24


Tabela 2 – Preporu% ene minimalne vrijednost indeksa pouzdanosti β (ULS) [10]Table 2 – Recommended minimum values of reliability index β (ULS) [10]

Minimum values for β Reliabilityclasses Reference period of 1 year Reference period of 50 years

RC3 5,2 4,3RC2 4,7 3,8

RC1 4,2 3,3

6 ZAKLJU"CI

Analiza gra'evinskih konstrukcija podrazumjevaviševarijabilne probleme, gdje su skoro sve varijablestohasti%ke. Prvi korak u analizi pouzdanosti je defi-nisanje funkcije gusto(e vjerovatno(e pojedinih varijabli.Postupci koji se primjenjuju svode se na upore'ivanjezakonitosti promjene varijable sa uobi%ajenim zakonito-stima koje se koriste u gra'evinarstvu, opisanim u radu.Pouzdanost definisanja zakonitosti je ve(a što je ve(abaza podataka provedenih istraživanja odre'ene vari-

able. Kod prefabrikovane gradnje pouzdanost prijesvega zavisi od pouzdanosti veza prefabrikovanih ele-menata. Realan mehanizam rada veze može se utvrditisamo eksperimentom. Stoga kod prefabrikovanih vezaostaje problem unificiranja postupaka analize. Pove(anjebaze podataka o ponašanju pojedinih montažnih vezadaje mogu(nost za primjenu analize pouzdanosti u ovompodru% ju gra'evinarstva. U tom smislu analiza pouzda-nosti prezentirana u ovom radu pove(ava postoje(ubazu statisti%kih podataka potrebnih za definisanjepouzdanosti prefabrikovanih veza. Analiza je ura'ena zaeksperimentom utvr 'ene srednje vrijednosti i varijacije,kao i za varijacije preporu%ene od autora koji su se bavilirazmatranom problematikom. Za analizu je usvojenanormalna raspodjela slu%ajno promjenljivih. Koeficijentikorelacije izme'u pojedinih slu%ajno promjenjivih suusvojeni za uslove eksperimenta. Zbog nedovoljnog broja ispitanih uzoraka za primjenu pouzdane statisti%keprocedure, koeficijenti korelacije su reducirani nao%ekivane minimalne vrijednosti. Usvojena je redukcijakoeficijenta korelacije

φφρρ S E S I C i ,, za 20%. Naime,

smatra se da slu%ajno promjenljive imaju jaku korelacijuako je njihov koeficijent korelacije ρ ≥ 0,50. Pošto jekorelacija momenta inercije i krutosti spoja, kao i krutostispoja i modula elasti%nosti betona, kod ispitanihmontažnih veza jaka, u sklopu diskusije su razmatranikoeficijenti korelacije umanjeni na vrijednost koja je nadonjoj granici dobre korelacije.

Prema tome usvojeni statisti%ki parametri su:48,0;72,0;4,0 ,,, === φφ ρρρ S E S I E I C C .

Varijacija stepena popustljivosti i moment nadosloncem su:

6 CONCLUSIONS

Analysis of civil structures includes multivariableproblems, where almost all stochastic variables. The firststep in analyzing the reliability is definition of probabilitydensity function of individual variables. The proceduresapplied are reduced to comparing the law of the variablechanges with the usual laws that are used in civilengineering, as described in the paper. Reliability ofdefinition of laws is greater the larger the databaseconducted research specific variables. By prefabricated

buildings the reliability mainly depends on the reliabilityof the precat connections. Realistic work mechanism ofthe connection can be established only in the experi-ment. Therefore, the precast connections remainsproblem of analysis procedures unification. Increase thedatabase on the behavior of individual precast con-nections gives the possibility for the application ofreliability analysis in this area of civil engineering. In thissense, the reliability analysis presented in this paperincreases the existing base of statistical data required fordefining the reliability of precast connections. Analysis isdone for experiment determined the mean and variation,as well as variations recommended by authors who havedealt with issues under consideration. For the analysisadopted a normal distribution of random variable. Cor-relation coefficients between individual random variableadopted for the experiment conditions. Due to an insuf-ficient number of tested samples for implementation ofreliable statistical procedure, the correlation coefficientswere reduced to the expected minimum values. Adoptedis to reduce correlation coefficients

φρ

S I ,and

φρ S E C , of

20%. It is considered that random variables have astrong correlation since the correlation coefficient is ρ ≥0,50. Since the correlation between moment of inertiaand connection stiffness, and connection stiffness andmodulus of elasticity of concrete strong for researchedprecast connection, in the discussions were consideredcorrelation coefficients reduced to a value that is at the

lower limit of good correlation. Accordingly, the correlation coefficients wereadopted: 48,0;72,0;4,0 ,,, === φφ ρρρ S E S I E I C C

Variation of degree of yielding and moment onsupport are:

( ) %03,29=αV

( ) %10%8 .exp.,. =


22/24


minimalne vrijednosti koeficijenata korelacije, je manjaod usvojene za uslove eksperimenta, Na osnovuprethodne analize preporu%uje se za potrebe definisanjapouzdanosti montažnih veza, koje se ponašaju sli%nokao ispitane predmetne montažne veze, koristiti sljede(eminimalne koeficijente korelacije:

− korelacija momenta inercije i modula elasti%nostibetona ρI,Ec = 0,40 ,

− korelacija momenta inercije i krutosti veze ρI,Sφ =0,72 ,− korelacija modula elasti%nosti betona i krutosti

veze ρEc, Sφ = 0,48 ,− korelacija širine i debljine elementa ρb, h = 0 Ova preporuka važi za analize pouzdanosti

konstrukcija za koje nisu provedena ili su u malomobimu provedena eksperimentalna istraživanja. Kodeksperimentalnih istraživanja ve(eg obima preporukamože biti konzervativna.

Analiza pouzdanosti, prezentirana u ovom radu,obuhvatila je nepouzdanosti mehani%kih veli%ina: optere-/enja, geometrije i svojstava materijala. Me'utim, pou-zdanost gra'evinske konstrukcije zavisi od svih u%esni-

ka u izgradnji objekta (investitor, projektant, izvo'a% inadzor). Za adekvatnu pouzdanost konstrukcije važne

su sve faze «stvaranja» objekta, što podrazumjeva:− kompetentnost lica ili tima za izradu projektnog

zadatka i tenderske dokumentacije;− odabir projektantske ku/e i izvo'a%ke firme sa

kvalifikacionom strukturom radnika, opremom ireferencama koje smanjuju mogu/nost grešaka;

− sistem kontrole kvalitete u pogonima firme koja sebavi proizvodnjom prefabrikovanih konstruktivnihelemenata;

− kompetentnost i stru%nost osoblja koje provodinadzor i kontrolu kvaliteta radova, organizacija ifunkcionalnost inspekcijskih službi;

− pogodnost i primjenjivost zakonske regulative iz

oblasti osiguranja kvaliteta, prava i obaveza svihu%esnika gra'en a, kaznenih mjera i efikasnosti njihoveprimjene.

Pored prethodno navedenog, socijalne prilike i stanjemorala u lokalnoj ili široj društvenoj zajednici tako'eimaju uticaja na pouzdanost objekata. Stoga je zapotpuno definisanje prihvatljive pouzdanosti gra'e-vinskog objekta neophodno analizirati sve navedeneparametre.

Napomena:

Drugi autor (R. Foli() je zahvalan na finansijskojpodršci od strane Ministarstva nauke Republike Srbije uokviru projekta TR 16018.

minimum correlation coefficient, is less than the adoptedfor the experiment conditions. Based on performedanalysis is recommended for defining the reliability of theprecast connection, which behave similarly asresearched precast connections, use the followingminimum correlation coefficient:

− correlation between the moment of inertia andmodulus of elasticity of concrete

− ρI,Ec = 0.40,− correlation between the moment of inertia andconnection stiffness ρI,Sφ = 0.72,

− correlation between the modulus of elasticity ofconcrete and connection stiffness

− ρEc, Sφ = 0.48,− correlation between width and thickness of the

element ρb, h = 0.This recommendation applies to the analysis of the

reliability of structures that are not implemented or are insmall scale experimental research was conducted. Withlarge scale experimental research recommendationsmay be conservative.

Reliability analysis, presented in this paper, include

the unreliability of mechanical quantities: load, geometryand material properties. However, the reliability of civilstructures depends on all participants in born of building(investor, designer, contractor and supervision). Foradequate reliability of structure all phase of "creating"building are important, which includes:

− competence of the person or team to create aproject task and tender documents;

− selection of design houses and contractor with theadequate company qualifications structure of workers,equipment and references, which reduces the possibilityof error

− a system of quality control facilities in the companythat manufactures prefabricated structures elements;;

− competence and expertise of personnel whosupervise and control the quality of work, organizationand function of inspection services

− suitability and applicability of the legislation in thefield of quality assurance, rights and obligations of allparticipants of construction, crime rate and efficiency oftheir application.

In addition to the foregoing, the social conditions andstate of moral in the local or wider community also havean impact on the reliability of buildings. Therefore, for thefull definition of acceptable reliability of buildings isnecessary to analyze all of these parameters.

Note:

The second author (R. Foli() is grateful for thefinancial support by the Ministry of Science of theRepublic of Serbia in the scope of the project TR 16018 .


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7 LITERATURAREFERENCES

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Basis of structural design, CEN, 2004.[11] Gomes H.M., Awruch A.M., Rocha M.M.: Reliabilityanalysis of reinforced concrete structures,Proceedings of 14th International Conference onStructural Mechanics in Reactor Technology(SmiRT 14),Lyon, France, August 1997., pp. 47-54.

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University Publishing House, 2001., pp. 20–25.[16] Kottegoda N.T., Rosso R.: Applied Statistics fo

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[18] Mehlhorn, G.: Schwing, H., Tragvehalten von ausFertigteilen zusammengesetzten Scheiben,Wilhelm Ernst&Sohn, Berlin 1977.

[19] Melchers, R.E.: Structural Reliability Analysis anPrediction, John Willey & Sons, Inc., New York,1999.

[20] Pham H.: Handbook of Reliability Engineering ,

Zbornik radova, XXIV. Kongres Društva zaispitivanje i istraživanje materijala i konstrukcijaSrbije, Div%ibare, Srbija, 15-17 October 2008, pp.273–282.

[21] Schnetgoeke R., Hosser D.: Application oreliability-based system assessment using a bridgeexample, Construction Informatics Digital Libraryhttp://itc.scix.net

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[23] Zenunovi( D., Foli( R.: Rigidity of Precast Plate –Monolithic Wall Connection, POLLACK PERIODICA, An International Journal for Engineering and Infor-mations Sciences, Vol.2, No.3, 2007, pp .85–96.

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[25] Kneževi( M., Zejak R.: Neuronske mreže –primjena za izradu prognoznog modelaeksperimentalnog istraživanja za vitke armirano-betonske stubove, Materijali i konstrukcije 51(2008) 1, Beograd, pp. 41–56.

http://itc.scix.net/http://itc.scix.net/


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REZIME

POUZDANOST AB MONTAŽNIH VEZA

Damir ZENUNOVI Radomir FOLI

U prvom dijelu rada date su teoretske osnove ii

pregled stanja aproksimativnih postupaka za analizupouzdanosti. Nadalje, u radu je prezentiran nastavakistraživanja veze montažne plo%e i monolitnog zida,obavljenih u periodu 2004.-2008. godina, koja suprovedena sa ciljem definisanja mehanizma radaovakvih veza. Rezultati istraživanja i preporuke autorakoji su provodili sli%na istraživanja korišteni su za analizupouzdanosti dobijenih rezultata. Za analizu vezemontažne plo%e i monolitnog zida korištena je metodadiskretnih ta%aka. Na osnovu provedene analize uzaklju%cima se daju preporuke za analizu sli%nih spojevau gra'evinskoj nauci i praksi.

Klju%ne re%i: analiza pouzdanosti, funkcija gusto(evjerovatno(e, indeks pouzdanosti, popust

Documents

Pouzdanost Ab Montažnih Veza