Gradjevinski Materijali i konstrukcije 3 2014

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ISSN 221 7-8 139 (Pr int) UDK: 06.055.2:62-03+620.1+624.001.5(497.1)=861ISSN 2334-0229 (Online)

2014. GODINA

LVII

GRA ĐEVINSKI MATERIJALI IKONSTRUKCIJE

BUILDING MATERIALS AND

STRUCTURES Č A S O P I S Z A I S T R A Ž I V A N J A U O B L A S T I M A T E R I J A L A I K O N S T 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

DRUŠTVO ZA ISPITIVANJE I ISTRAŽIVANJE MATERIJALA I KONSTRUKCIJA SRBIJE

SOCIETY FOR MATERIALS AND STRUCTURES TESTING OF SERBIA

DDIIMMKK

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Odlukom Skupštine Društva za ispitivanje materijala ikonstrukcija, održane 19. aprila 2011. godine u Beogradu,promenjeno je ime časopisa Materijali i konstrukcije i od sadaće se časopis publikovati pod imenom Građevinski materijali ikonstrukcije.

According to the decision of the Assembly of the Society forTesting Materials and Structures, 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







GRA ĐEVINSKI MATERIJALI I KONSTRUKCIJE 57 (2014) 3 (3-20) BUILDING MATERIALS AND STRUCTURES 57 (2014) 3 (3-20)

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EUROCODE 8: USE OF ADVANTAGEOUS FORMULATIONS FOR IMPROVED ANDSAFE DESIGN

EVROKOD 8: UPOTREBA POVOLJNE FORMULACIJE ZA POBOLJŠANO I SIGURNOPROJEKTOVANJE

Zdravko BONEVStanislav DOSPEVSKI

ORIGINALNI NAUČNI RAD

ORIGINAL SCIENTIFIC PAPERUDK: 624.042.7:006.44EN

1 INTRODUCTION

In the recent years current design philosophiesevidently are in the period of evolution. The generalpurpose considering seismic resistant design is to savehuman live and minimize human damages inengineering structures. The experience from the pastearthquakes shows that sometimes design theory is notin compliance with the observed results. A lot of effortshave been directed towards achievement a goodcorrelation between theoretical background and

observation results. The use of structural mechanicsprinciples contributes more or less towards improvingthe precision of computational procedures. This is goodbackground for implementation of performance-basedseismic design. More adequate approaches and betternumerical simulation in the field of seismic action andefficient dynamic models of structures are developed.

EN 1998-1 provisions are great contributes towardsvery good agreement between theory and practice.Implementation of capacity design principles is themilestone for better predictability of the plasticmechanisms when structures are subjected to seismicground motion. Structures are designed usingperformance based seismic design methodology, which

allows for better identification of structural performance.European standard proposes very good balancebetween reliability of the methods and simplicity of theiruse. EN 1998-1, [9] provides better protection of humanlife and civil engineering structures as well.

Professor Zdravko Bonev, PhD, University of Architecture,Civil Engineering and Geodesy, Department of StructuralMechanics, 1 Hristo Smirnenski Blvd, Sofia, Bulgaria,e-mail: [email protected] Dospevski, PhD Civil Engineer, University of

Architecture, Civil Engineering and Geodesy, Department ofStructural Mechanics, 1 Hristo Smirnenski Blvd, Sofia,Bulgaria, e-mail: [email protected]

Figure 1 illustrates the main concept in developmentstrategy for decreasing the gap between theory andpractice. At first, design theory should become moreprecise, using fewer assumptions and more refinedmethods, which are capable to capture the mostimportant features of seismic performance of structures.The “old” generation of BG design codes (solid line) isreplaced with a new expanded figure (dashed line)showing new advantageous level of the design theory.

On the other hand construction methodology is alsoexpanded, moving from old (solid line) to new (dashedline) position. The existing gap between figures denotedby solid lines and dashed lines illustrates the potentialdistinctions that may appear between theory andpractice. Expansion of design methods in EN 1998-1 isachieved by implementation of more science into thedesign procedures, for instance nonlinear seismicperformance, probabilistic methods in defining the limitstates, use of more sophisticated models for seismicaction. Performance requirements and detailing rules ofEN 1998-1 (dashed line) can be pointed out as a matterof new civil engineering practice.

As a result of comparison it is evident that the gap

between dashed line figures is going to be smaller thanthe gap between solid line figures. The basic idea of EN1998-1 is to implement more scientific developments intothe new theoretical background and to carry outcorresponding changes in practical regulations which arein conformity with the new design principles. Forexample considerable attention is paid to ductility as abasic tool for energy dissipation and design loadreduction. Special detailing rules are proposed to ensureductility in the practice as a property of structural criticalzones and connections.



GRA ĐEVINSKI MATERIJALI I KONSTRUKCIJE 57 (2014) 3 (3-20)BUILDING MATERIALS AND STRUCTURES 57 (2014) 3 (3-20)

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The subject of the paper is closely related to the“open topics”, which are the subject of the National Annex and Nationally determined parameters for Bulgaria. The main goal of the paper is to share theexperience of the team, involved to carry out the workrelated to Part 6: Masts, Chimneys and Towers and Part3: Repairing and Strengthening of Reinforced Concreteand Masonry Structures.

2 SELECTED TOPICS OPENED FOR DISCUSSION

2.1 Topic 1: Accounting for spatial variability ofseismic action

For a number of structures seismic ground accelera-tions are assumed to be uniformly distributed at the baseof the structure similarly to “shaking table effect“. For irregular or long in plan structures like storages, bridgesand tubes uniform distribution of the ground accelera-tions or displacements is no longer applied. EN 1998-1

states that spatial variability of the seismic action shouldbe accounted for. For tall and slender structures spatialvariability of the seismic action on input also should betaken into consideration. There are mainly two modelswhich are currently in use to do this. First model impliesthe ground acceleration distribution at the base to followthe shape of the “standing wave function“, typical forshear waves and surface waves of Love. The second

model proposes to enter as seismic input into thestructure both translational components of groundaccelerations and rotational components of the groundaccelerations.

Model 1: ground accelerations which follow theshape of standing wave function

(References [3], [4], [7], [8], [15], [18], [19], [20] an[21])

In this case seismic input is represented byhorizontal ground accelerations, which are evidently,non-uniformly distributed. Figure 2 shows simple ground

Figure 1. General strategy for compliance between design methods and engineering practice

Figure 2. Long in plan structure (horizontal plane XY) subjected to standing wave (shear wave)




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motion being consistent of accelerations which are non-uniformly distributed. A bridge structure is subjected tothis seismic input. The transferred motion of all massesattached to the deck structure, is the same as the motionof the base. Figure 2. shows that the groundaccelerations given in it provoke domination of torsionalresponse over the translation-based solution.

Figure 3 shows long in plan structure subjected tohorizontal accelerations at the base which are non-uniformly distributed. The most widely used model of seismic action requires however uniform distribution of the ground accelerations. This idea can be achieved byseparation of the long in plan building into separateparts, which behave as dynamically independent units.Ground acceleration function is also separated (dashedline) and each line part-by-part becomes easier to beapproximated with uniform line. Having the seismic jointsthe spatial effect of seismic loading now is reduced.Some recent investigations show that dynamicindependence of the separate units remainsquestionable because they interact through soil.

Model 2: using rotational components of the seismic ground motion (References [2], [10] and [22])

This model is provided in [10]. Numerical responseevaluation using [10]-regulations and approach is givenin [2]. The existing model of seismic action, having threetranslational components, is extended by adding the rota-

tion components of the ground accelerations as shownin Figure 4. The corresponding rotational SDOF systemdefining rotational motion is shown in the same figure.Thus, the full set of ground acceleration componentscontains six components – three translational and threerotational components (Fig. 4).

The elastic response spectra according to [9] of

Types 1 and 2 are shown in Figure 5 and they arerelated to translational accelerations measured on thefree field.

The definition of the rotational components of theground motion using response spectra facilities is carriedout in [10]. Following the pattern in Part 6 three rotationalspectra are introduced: first two spectra, defined byequation (1) are related with surface waves of Rayleighand imply rotational motion around both horizontaldirections. Third rotational spectrum in (1) is related torotational motion around the vertical axis and appears tobe a result of the surface waves of Love.

( )

( )

( )

1,7

1,7

2,0

e

x

S

e

y

S

e

z

S

S T S

V T

S T S

V T

S T S

V T

θ

θ

θ

π

π

π

=

=

=

(1)

Figure 3. Nonuniform distribution of horizontal ground accelerations at the base of the building.Use of seismic joints to mitigate the effect of torsional ground acceleration input

Figure 4. Full set of acceleration components at a point. Definition of acceleration spectrumthrough single rotational degree of freedom system




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It is seen that all rotational spectra are expressed interms of the elastic response spectrum, plotted in Figure5. All spectra in Equations (1) are parametricallydependent on shear wave velocity

S V and on the current

period of free undamped vibrations T . When the periodT tends to zero, all formulae in (1) produce infinitelylarge rotational accelerations. This tendency wasobserved and studied by M. D. Trifunac and Lee

considering records of rotational accelerations for specific earthquakes.

The plot of the rotational response spectra accordingto [10] and in conformity with Equations (1) arerepresented in Figure 6. Looking at both types of rotational spectra as a result of surface waves (Rayleighand Love waves) as a tendency it can be seen that highfrequency (small periods) response goes to infinity andor may become quite large for all ground types. Stiff soils(Ground types A and B) show narrow spectralamplification band, whereas soft soils (Ground types Cand D) have more wide spectral amplification band. Itcan be expected that only stiff structures (with smallfundamental period) can be affected essentially by

rotational accelerations. On the other hand, spectralaccelerations in large period range ( T larger than 1.0 sfor instance) are very small and their contribution can beneglected.

For MDOF systems response spectrum method isalso applicable. To do this the application of this methodincluding translational components only should beextended by implementation of rotational components,too, see Figure 6.

Application of the response spectrum method totower structure is discussed in the following issue.Numerical model is prepared on the basis of finite

elements – 3D or SHELL elements, as shown in Figure7. The numerical model, composed by SHELL elementsallows for calculation of cross section shapedeformations, but the number of dynamic degrees offreedom is increased very much (Figure 7).

The transferred translational and rotational motion ispresented in Figure 8. Translational motion ischaracterized by constant distribution of transferredaccelerations in elevation. Rotational motion ischaracterized by linear distribution of the transferredmotion induced by rotational accelerations of the base.

Translational and rotational transferred motions areconsidered and discussed below in order to determinethe design seismic forces and the most importantparameters for further calculations.

Because of the nature of response spectrum method,two separate and independent analyses are carried out.First one deals with the action effect, obtained only as a

Figure 5. Elastic response spectrum: Type 1 (left) and Type 2 (right)

Figure 6. Elastic response spectra for rotational accelerations: spectra, obtained as a result of Rayleigh waves (left)and spectra obtained on the basis of Love waves (right)




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Figure 7. 3D frame elements and shell elements used to compose the dynamic model of a tower.Shell elements allow for catching cross section deformations

Figure 8. Transferred motion of the ground and its distribution in elevation(left – translational ground motion, right – rotational ground motion as a result of surface waves)

result of the translational component. This kind of analysis is traditional type of analysis well known from

seismic resistant design courses. The second analysisdeals with the structure subjected to rotationalcomponent only. By making use of componentcombination rule action effects from both analyses canbe combined to account for that components are actingsimultaneously.

Details concerning all calculations and notations aregiven in Appendix A. Only final results and simplenumerical example are discussed herein.

A simple cantilever structure – tower is shown as aplane structure in Figure 8. The structure is subjected tohorizontal ground motion, denoted by superscript X androtational ground motion, denoted by superscript θ . Both

components are acting simultaneously so coupled effectin all responses should be accounted for. The dynamic

model has twenty masses and twenty degrees of freedom. structure is subjected to translational ground accelera-tion components only

The solution of this problem can be found elsewherein current courses on structural dynamics andearthquake engineering. The only modal values of baseshear force

,

x

i baseV (Equation (2)) and base flexural

moment,

x

i base M (Equation (3)) are considered. All

quantities entering in both equations are explained anddiscussed in Appendix A.




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( ) ( )2

*

,

x x

i base i i e iV m S T = Γ (2)

( )*

,

x x

i base i i i e im hS T θ = Γ Γ (3)

Since modal correlation exists between two modalpeaks of each modal pair, this is taken into account by

correlation coefficients of Der Kiureghian (Appendix A).Thus the peak values of the base shear force and baseflexural moment are determined after application of CQCas modal combination rule.

, ,

x x x

base ij i base j base

i j

V V V ρ = ∑∑ (4)

, ,

x x x

base ij i base j base

i j

M M M ρ = ∑∑ (5)

It is worth noting that the value of,

x

i baseV obviously

always remains positive (see Equation (2)), whereas,

x

i base M may have either positive or negative value (see

Equation (3)).

structure is subjected to rotational ground accelerationcomponents only

The solution of this problem should be found inconformity with response spectrum method and theory.Only modal values of base shear force

,i baseV θ (Equation

(6)) and base flexural moment,i base M θ (Equation (7))

are considered. All quantities entering in both equationsare explained and discussed in Appendix A.

( )*

,

x

i base i i i e iV m hS T θ θ = Γ Γ (6)

( ) ( )2

* 2

,i base i i e im h S T θ θ = Γ (7)

It is worth noting that the value of ,i baseV θ may have

positive or negative sign (see Equation (6)). The value of

,i base M θ ever remains positive.

The need for modal combination arises because acorrelation can be found between each arbitrary pair ofmodal peaks. This is carried out using correlation

coefficients of Der Kiureghian (Appendix A). Thus thepeak values of the base shear force and base flexuralmoment are determined after application of CQC asmodal combination rule.

, ,base ij i base j base

i j

V V V θ θ θ ρ = ∑∑ (8)

, ,base ij i base j base

i j

M M M θ θ θ ρ = ∑∑ (9)

peak action effects as a result of simultaneous action ofboth components of the ground accelerations

Now the fact that both components of the seismicground motion (accelerations) are acting simultaneously,should be taken into account. It was proven that thecross correlation factors for translational and rotationalacceleration records are zero. This shows that nocorrelation between translational and rotational

component exists and peak values of base shear forcemax baseV and flexural moment max base M when both

components of the seismic ground motion are actingsimultaneously, can be found using SRSS as componentcombination rule.

( ) ( )2 2

max x

base base baseV V V θ = + (10)

( ) ( )2 2

max x

base base base M M M θ = + (11)

Note that Equations (9) and (10) are usinginformation after the peak responses due to each of theground motion component have already beendetermined.

effective mass ratios

Effective mass ratios point out whether the numberof modes taken into account into the analysis is sufficientfrom the viewpoint of accuracy. For translational motionthis problem is already settled and used for a number ofyears. Calculations are performed using the resultprovided with Equation (12).

For rotational ground motion new definitions for thetotal mass

Tot M θ and for the modal participation factor

iθ Γ are implemented (see Appendix A) correspondingly.

Calculation of the effective mass ratio is performed usingthe result (13):

( )2

*

1 0, 90 0, 95

N x

i i x i

x

Tot

m

M α =

Γ= = ÷

∑ (12)

( )2

*

1 0, 90 0, 95

N

i i

i

Tot

m

M

θ

θ

θ α =

Γ= = ÷

∑ (13)

Information about all quantities entering in Equations(12) and (13) is provided in Appendix A.

Figure 9 illustrates the relationship “effective massratio versus number of modes, included in the analysis”.Solid line represents the case of horizontal motion only(Equation (12)), whereas dashed line shows thevariation of effective mass ratio in case of rotationalmotion of the base. It is seen that in the case oftranslational component only 3 modes are sufficient toachieve at least 90% accuracy. In the case of rotationalcomponent (dashed line) only one mode is sufficient toachieve 97% accuracy. This can be explained by thefact, that the shape of the rotational transferred motion




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Figure 9. Sufficiency of the number of modes implying translational only (solid line)and rotational motion only (dashed line)

Figure 10. Elastic acceleration spectra: for translational motion (left) and for rotational motion - Rayleigh waves (right)

(see Figure 8) can be approximated accurate enoughusing only first (fundamental) mode of natural vibrations.It means that for tall and slender structures like towersthe only one mode of natural vibrations is sufficient toreproduce the effect of base rotation. High moderesponses are unaffected by the rotational motion of thebase.

Numerical example

EN 1998-6 requires structures higher than 80 meters

to be examined to both translational and rotationalcomponents of the ground motion. Supplemental

condition is 0,25 g a S g > ( g a is the design ground

acceleration, S is the soil coefficient and g is the

acceleration of gravity). In the considered example theresponse of an existing tower is studied for which theheight is 60 m (less than 80 m)and 0, 27 0, 25 g a S g g = > . Damping ratio as a

percentage of the critical is 5%ξ = . Elastic response

spectrum and Bulgarian National parameters are used tocalculate and plot the spectra, needed for calculationsand representing the seismic input. Figure 10 illustrates

graphically these data. Ground type C is used incalculations.

Figure 11 shows graphically the relationship “basemoment - included number of modes”. Solid linerepresents the case in which both componentshorizontal and rotational are acting simultaneously.Dashed line shows the variation of the base momentwhen only horizontal component of the seismic action isconsidered. It is seen that, as expected, three modes aresufficient to obtain both lines. Obviously, high frequencyresponse has no essential contribution over the basemoment value. It is seen also, that taking into accounthorizontal and rotational components of the groundmotion produces larger response than the responseincluding only translational component. Dashed areagives impression about the distinction between twoseismic action models. Difference is 11%, which is noton the side of safety when rotational component ofground accelerations is omitted.

It is shown (figure 9) that fundamental period is themost important parameter which forms the response dueto rotational component. EN 1998-1 has a requirementfor sufficient stiffness aiming to ensure limitations for thedisplacements. As a result the fundamental period is




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going to be small (around zero) which leads to largerotational component contribution (see Figure 6). In theperiod (0. - 1.0 s) large contribution of rotational compo-nent is expected because structures should be designedto be stiff enough. This zone may generate amplificationof the response because of the large influence of rotational component. The opposite opportunity is when

structures have insufficient stiffness and are flexible. Inthis case it is expected the displacements to grow upand exceed the prescribed limits. Within the large periodrange (2.0 – 3.5 s) the horizontal component is expectedto contain long period component which may lead toresponse amplification.

Within the period range (1.0 s - 2,0 s) large dynamicamplification is unlikely to expect. Figure 12 showsproperly the three zones. The large amplification effectof first and third zones can be reduced by using viscousdampers and seismic protection technics. Theamplification effect of the first zone can be reduced byhigh frequency component of the viscous damping. The

amplification effect of long period zone can be reducedby low frequency component of the viscous damping.

Figure 11. Peak responses for the base moment: including translational and rotational componentssimultaneously (solid line) and translational component of the ground acceleration acting alone (dashed line)

Figure 12. Suggestion for mitigation of amplification by making use of added damping being implementedas equivalent viscous damping




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2.2 Topic 2: Performance based seismic designand need verification for strengthening ofexisting structures (References [5], [11], [12],[14], [16], [17] and EN 1998-3)

Part 3 (EN 1998-3) is very ambitious attempt tocreate new regulations for design of strengthenedstructures. It is compulsory to proof two importantissues: 1) the need for strengthening, and 2) the benefitof strengthening. The study discussed in the paper proposes numerical technique, which allows for provingthe need and the benefit of strengthening. Both proofsare carried out on the basis of adequate numericalmodels, which are capable of caching inelasticbehaviour and effects of it. Another important factor –the cost of the strengthening and economic viability. Thequestion that is able to afford strengthening is dependenton governmental priorities and should be settled bygovernmental institutions. In this topic 2 only the needand benefit are considered.

Capacity design principles and rules are widely used

in [9] to improve the seismic performance of existingbuildings designed in conformity with old generation of codes. This can be done by doing the following 1)improvement of the energy dissipation capability andductile performance; 2) improvement of the predictabilityof the plastic and failure mechanisms. Energy dissipationand ductile behaviour lead to reduction of design seismicforces. Prediction of the plastic and failure mechanismsis a great advantage from the viewpoint of methodology.The basic idea is to control structural performance bymore adequate regulations and more proper designphilosophy.

For new designed structures the use of capacitydesign rules, performance requirements and detailing

rules of [9] contribute towards achievement of globalplastic mechanism (Figure 14). Capacity designphilosophy allows for the plastic mechanism to bepredicted and implemented in calculations. Advantagessuch as good energy dissipation distribution in elevation,elastic and strong columns and dissipative beamscontribute towards good performance for local andglobal ductility. Another advantage of new design

principles is that the influence of P - ∆ effect is limitedbecause there is no storey with strongly dominatinginterstorey drift over the other drifts.

The performance based evaluation procedureimplies the use of two types of capacity spectra: actualcapacity spectrum (ACS) and target capacity spectrum

(TCS). Seismic demands of the action effects are easyobtained as a result of design seismic loading. Thealgorithm proposes comparison of seismic demandsbefore and after strengthening. The analysis results areeasy to be interpreted and provide argumentation for strengthening and benefit of it. Comparable analysis of the results before and after strengthening is a goodbasis for making conclusions by design engineers.

Some basic terms and formulations are discussedbelow.

actual capacity spectra (ACS)

Actual capacity spectrum corresponds to a certainstructure, which is designed according to EN 1998-1 or

according to some other seismic resistant design code. Actual capacity spectra however can be determined alsofor structures which are designed without proper designcriteria and performance requirements. At first baseshear force – roof displacement diagram (pushovercurve) is calculated, then on the basis of simplemodifications actual capacity spectrum (ACS) isobtained. This spectrum is a plot of “first flooracceleration against modified displacement”. Modified

displacement*

u is related to the unique displacement ofequivalent single degree of freedom (ESDOF) system.“Zero length” plastic hinge concept is applied. Detailsconcerning plastic hinge parameters are not discussedin the paper. As a matter of fact, it is clear that actual

capacity spectrum is obtained on structural data.Figure 13a illustrates one three storey three baysreinforced or steel frame. It is loaded statically withmonotonically increased vertical loads first and afterending of the process a new loading case withmonotonically increased horizontal loads is initiated. Action effects are obtained as a sum of the action effectsfrom both subsequent loading cases. Lateral loads areapplied at the storey levels and their increase continuestill the base shear capacity is reached.

The typical force-deformation curve is shown inFigure 13b. As a matter of fact further theoreticaldevelopment is performed using idealized bi-linearrelationship. This line is obtained on the basis of actualrelationship “base shear force – roof displacement” usingenergy balance principle recommended [9]. Howeverthis solution is not a unique one and it should be takeninto account that it becomes a source of numerical error.

Figures 14 and 15 illustrate global (G) and “soft firststorey” (F) mechanisms which propose differentdisposition of the plastic hinges. In first case plastichinges are located at the end of the beams when loadingprocedure is still not terminated. Dissipated energy hasrelatively uniform distribution in elevation; in the secondcase dissipated energy is concentrated only in the firststorey. This implies that failure mode is a product ofinelastic deformations of the first storey only.

In old generation of codes capacity design principlesdo not take place in seismic resistance calculations. The

prediction of the plastic mechanism in the process ofdesign is impossible. In many cases plastic mechanismmay surprise the design engineer and may haveunexpected shape. Among all feasible mechanisms themost dangerous and unacceptable is the mechanism“soft first storey“ (F - mechanism). For the sake of safetythis mechanism is used in checking procedure, seeFigure 15. This approach is applied when existingbuilding is studied and capacity design rules are notused.




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Figure 13. Determination of actual capacity spectrum:a – loading patterns for vertical and horizontal forces used in static pushover analysis;

b – pushover curves: actual (dashed line) obtained through computations and idealized (solid line)bi - linear curve suggested in theoretical development.

Figure 14. Global (G) plastic mechanism




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Figure 15. Soft first storey (F) plastic mechanism (local mechanism)

Appendix B contains information about transfor-mation of MDOF structure and its system of differentialequations into ESDOF (equivalent single degree offreedom system) equation (see Figure 13a) assuming Gmechanism from Figure 14. The same transform for F -

mechanism in Figure 15 mathematically seems to be aparticular case of the more general case of Gmechanism provided in Annex B. The introduction of Gand F mechanisms implies the following explanations.

F – mechanism

The vector of dimensionless lateral displacements forF - mechanism takes very simple form containing onlyunity components:

1

1

1

⎧ ⎫⎪ ⎪⎨ ⎬⎪ ⎪

⎩ ⎭

Φ = (14)

Following Appendix B and taking into account Eq.(14) the quantities mentioned there can be calculated togive

mass of the ESDOF system:

[ ] 3

*

1

T

i i

i

m m m mν =

= Φ = Φ =∑ (15)

mass parameter

[ ] 3

2

1

ˆ T

i i

i

m m m m=

= Φ Φ = Φ =∑ (16)

participation factor ˆ 1Γ = (17)

resisting force of the ESDOF system

3

*

1

T

i i

i

f f f V =

= Φ = Φ =∑ (18)

equation of the motion of ESDOF system:

( ) g

V u t u

m+ = − (19)

It is worth noting that the basic force –displacement relationship in the case of Fmechanism is “base shear force V – roofdisplacement u ”. Because of Eq. (17) thedisplacement of the ESDOF system is equal to the roofdisplacement in MDOF system. The peak response ofESDOF system is easy expressed using responsespectra.

G – mechanism

In the case of G mechanism the mathematicaltransform MDOF – ESDOF system is shown in AppendixB. This kind of transform is needed because the peakresponse of ESDOF system is easy evaluated throughresponse spectra.

target capacity spectrum (TCS)

Target capacity spectrum (TCS) should comply withthe performance requirements in largest extent. It shouldcover deformation criteria such as limited elastic

displacement*

yu for G mechanism and limited first

storey elastic drift yu , ductility demand should be less

than ductility capacity (basic requirement of [9]). For Fmechanism the elastic deformation criterion can bewritten in the form (see Figure 16). :

250 y

H u ≤ (19a)

where H is the total height of the building. The ESDOFsystem should have prescribed values for thedisplacement ductility demand, behaviour factor.Prescribed values are entered by the design engineer asinput.

Determination of target capacity spectrum is ageometric problem which is settled by iterations. Allbasic parameters are varying in previously specifiedlimits in order to satisfy simultaneously all requirementsin largest extent. In contrast with actual capacityspectrum the procedure of determination of the targetcapacity spectrum does not require any structural data, it just couples specified design regulations. Figures 16 and17 are graphically illustrating the determination of the




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target capacity spectrum (TCS) - Figure 16 for Fmechanism and Figure 17 - for G - mechanism cor-respondingly.

Calculation procedure assuming F mechanism canbe generalized in the following steps (Figure 16). Thecalculation process is iterative and the results areapproximate. The iterative algorithm can be representedusing the following steps:

1. Specify yu using equality sign in Eq. 19a asupper limit for yielding displacement.

2. Calculate the ductility demand on the basis of theposition of point PP.

3. Find the point EPP (elastic performance point)basing on the observation of Newmark for equaldisplacements.

4. Find the elastic (initial) stiffness for ESDOF systemconnecting EPP with the origin of the coordinate system.

5. Find the acceleration at yielding / yV m ( m - total

mass of the structure).6. Find the behaviour facto q r as a ratio of (ordinate

of EPP)/(ordinate of PP).If the results from items 2), 3) 4), 4) and 5) do not

comply with the prescribed values of same quantitiesthen reduce yu and go back to items 2), 3), 4), 5) and 6).

For G - mechanism(see Figure 17) the iterativealgorithm is remaining the same but all quantities shouldbe related to the equivalent single degree of freedomsystem described mathematically in Appendix B havingthe upper superscript ( * ).

Figure 16. Determination of target capacity spectrum using base shear force – roof displacement relationship. Dashedline represents the elastic demand spectrum, solid line represents the design demand spectrum for accelerations usingspecified ductility, and dark solid line is the target capacity spectrum

Figure 17. Determination of target capacity spectrum using effective force – effective displacement relationship. Dashedline represents the elastic demand spectrum, solid line represents the design demand spectrum for accelerations using

specified ductility, and dark solid line is the target capacity spectrum






16

Let us assume that a simple building designedaccording to code of old generation is analyzed. Theactual capacity spectrum of this structure withoutstrengthening is ACS – 1. Let us imagine that the targetcapacity spectrum TCS looks like this shown in Figure19. Comparison between ACS – 1 and TCS illustratesthat the structure needs greater shear force capacity towithstand the design earthquake of [9]. The existing

structure shows insufficient initial elastic stiffness andinsufficient yield strength. This means that the necessityfor strengthening is evident. After strengthening structureshows an improved capacity and it is capable ofwithstanding of the design seismic action. The actualcapacity curve ACS – 2 has an excessive capacity withrespect to TCS line and level of safety is increased dueto the strengthening. Excessive elastic stiffness andexcessive yield strength are also evident consideringboth ACS – 2 and TCS lines.

Looking at Figure 20 after comparison betweenidealized capacity spectra TCS and ACS one mayconclude that the structure studied has insufficient

elastic stiffness and insufficient yield strength. Thus thisstructure needs strengthening. Figure 21 shows threedifferent opportunities for strengthening. First oneimplies strengthening by using fibre reinforced polymers- FRP. Elastic stiffness and yield strength are increasedby increasing the confinement. Second method impliesthe confinement to be implemented by R/C jackets. The

third opportunity is based on inverted steel “V ” – frame

implementation into the R/C beams and columns subassemblages.

Performance base seismic design method isapplicable only for low rise building structures. Onlysingle mode response is taken into account into thismethodology and limitations are considered asconsequence of this. For high-rise and tall buildingstructures multi-modal methods are recommendedbecause high frequency response may contributeessentially on overall response.

Figure 19. Comparison between TCS and ACS capacity spectra. ACS–1 represents the actual capacity spectrum of existing structure before strengthening.

ACS–2 is the actual capacity spectrum after strengthening the existing structure.

a b c

Figure 20. Different strengthening strategies and techniques:a – improved confinement using FRP (fibre reinforced polymers), [14];

b – use of R/C jackets, [14]; c – use of internally implemented steel framework, [1].




17

3 CONCLUSIONS

The following conclusions are made as a result ofproposed studies:

topic 1:

1. Inclusion of the rotational components results inlarger action effects and unfavourable results.

2. Inclusion of rotational components leads tomagnification of low period’s response.

3. An efficient peak response reduction can beachieved by making use of supplemental stiffnessproportional viscous damping.

topic 2 :

1. Comparable analysis between target capacityspectrum (TCS) and actual capacity spectrum of existing

structure before strengthening (ACS 1) is capable ofindicating whether strengthening is necessary or not.

2. Comparable analysis between target capacityspectrum (TCS) and actual capacity spectrum (ACS 2)of strengthened structure can be used to prove thebenefit and efficiency of strengthening.

3. Target capacity spectrum (TCS) is the spectrumwhich satisfies the Eurocode 8 provisions. The scatter

between TCS and ACS 1 shows how the parameters ofexisting structure should be changed. The scatterbetween TCS and ACS 2 can be used for prediction oflevel of safety after strengthening.

ACKNOWLEDGEMENTS

The encouragement and help of Professor emeritusRadomir Folic are greatly acknowledged by the authors.

4 REFERENCES

[1] Abdullah. Seismic Strengthening of RC BuildingStructurer. FYPE Japan Co., Ltd. (PPTpresentation).

[2] Bonev Z., E. Vaseva, D. Blagov and K. Mladenov.Seismic Design of Slender Structures IncludingRotational Components of the Ground Acceleration – Eurocode 8 Approach. Proceedings of the 14thEuropean Conference on Earthquake Engineering,30 August – 03 September, 2010, Ohrid,Macedonia, Paper 1793.

[3] Bonev Z., S. Dimova and P. Petrov. Study of theSeismic Design of Irregular Structural Systems.

European Association of Earthquake Engineering,Task Group 8: Asymmetric and Irregular Structures, Volume 1: Irregular Structures, EditorsF. Karadogan and A. Rutenberg, October 1999,Istanbul, pp. 57-71.

[4] Bonev Z. and A. Taushanov. Seismic Design of Structures According to Response SpectrumMethod coupled with Linear Analysis. IntegraEngineering, 2006 (book, in Bulgarian).

[5] Calvi Gian Michele, Stefano Pampann, P. Fajfar,M. Dolsek . New Methods for Assessment andDesign of Structures in Seismic Zones: PresentState and Research Needs.

[6] Chopra A. K. and Rakesh K. Goel. Capacity-Demand-Diagram Methods for Estimating SeismicDeformation of Inelastic Structures: SDF systems.Pacific Earthquake Engineering Research Center,PEER, 1999/02, April 1999.

[7] Dimova S., P. Petrov and Z. Bonev. Precising of the Design Seismic Loading of Irregular Structures.Proceedings of the 3rd Japan-Turkey Workshop onEarthquake Engineering, Volume 1, Istanbul, 2000,pp. 63-77.

[8] Dimova S., Z. Bonev and P. Petrov. SeismicDesign of Irregular Structures Including RotationalComponent of the Seismic Excitation. European Association of Earthquake Engineering, TaskGroup 8: Asymmetric and Irregular Structures,Volume 1: Irregular Structures, Editors F.Karadogan and A. Rutenberg, October 1999,Istanbul, pp. 73-87.

[9] EN 1998-1 Eurocode 8 : Design of Structures forEarthquake Resistance – General Rules, Seismic Actions and Rules for Buildings.

[10] EN 1998-6 Eurocode 8 : Masts, Chimneys and

Towers[11] Fajfar P. A Nonlinear Analysis Method forPerformance Based Seismic Design. EarthquakeSpectra, Vol. 16, No 3, pp. 573-592, August 2000.

[12] Fajfar P. Structural Analysis in EarthquakeEngineering – a Breakthrough of Simplified Non-Linear Methods. Published by Elsevier ScienceLtd., 12th European Conference on EarthquakeEngineering, Paper reference 843.

[13] Fajfar P., D. Marusic, I. Perus. The N2 Method for Asymmetric Buildings. First European Conferenceon Earthquake Engineering and Seismology,Geneva, Switzerland, 3-8 September, 2006, PaperNumber 539

[14] Fardis M. Seismic Design, Assessment andRetrofitting of Concrete Buildings, Based on ENEurocode 8, Springer Science+Business MediaB.V. 2009.

[15] Kuesel T. R. Earthquake Design Criteria ofSubways. Journal of the Structural Division,Proceedings of the American Society of CivilEngineers, Vol. ST 6, June 1969, pp. 1213-1231.




18

[16] Mestrovic G., D. Gizmar and M. Pende. Nonlinear Analysis of Structures According to New EuropeanDesign Code. Proceedings of the 14th WorldConference on Earthquake Engineering, October 12-17, 2008, Beijing, China.

[17] Static Pushover Analysis in Z_Soil PC & andrelated EUROCODE 8 regulations R_070202,January 2007.

[18] St John C. M. and T. F. Zahrah. Aseismic Designof Underground Structures, FEATURES SECTION:Seismic Effect on Underground Structures.Tunneling and Underground Space Technology,Vol. 2, Number 2, 1987, pp. 165-197.

[19] Tzenov L. Basics of Earthquake Engineering.Professor Marin Drinov Academic PublishingHouse, Sofia, 2002.

[20] Tzenov L., S. Dimova, Z. Bonev and P. Petrov.Seismic Resistant Design of Irregular Structures:Generalized Method for Determination of DesignSeismic Loading, on issue of Turkish EarthquakeFoundation, Technical Report TDV/TR 037-62,Istanbul Technical University, 2001.

[21] Todorovska M. and M.D. Trifunac. ResponseSpectra for Differential Motion of Columns,

Proceedings of the 12-th World Conference onEarthquake Engineering, 2000, paper 2638. [22] Todorovska M. Some Interferometry of Soil –

Structure Interaction Model with CoupledHorizontal and Eocking Response, Bulletin of theSeismological Society of America, Vol. 99, No. 2A,pp. 611-625, April 2009.

SUMMARY

EUROCODE 8: USE OF ADVANTAGEOUSFORMULATIONS FOR IMPROVED AND SAFEDESIGN


The purpose of the paper is to share the experiencein application of Eurocode 8. Following the existingframework of the European standard two engineeringdevelopment of two specific topics are proposed. Firstdeals with modelling the seismic action aiming to takeinto account spatial variability of the seismic action. It is

proven that the inclusion of rotational components mayinfluence the action effects and the results provideunsafe design. Evaluation is carried out throughresponse spectrum method. The second topic studiesthe use of performance based seismic design as a toolto prove the need for seismic strengthening andverification of the benefit. Both target and actual capacityspectra are compared to make assessment of the lateralresistance of existing and strengthened structure.Insufficient elastic stiffness and insufficient yield strengthare considered as the most typical situations frequentlymet in the design practice and which can be settled byseismic strengthening. Discussions and conclusionsconcerning the philosophy of the assessment procedureare presented.

Key words: Seismic resistant design to Eurocode 8,taking into account spatial variability of seismic action,performance based seismic design

REZIME

EVROKOD 8: UPOTREBA POVOLJNEFORMULACIJE ZA POBOLJŠANO I SIGURNOPROJEKTOVANJE


Svrha ovog rada jeste da se prenesu i podeleiskustva iz primene Evrokoda 8 (EN 1998). Sledećipostojeći okvir Evropskih standarda, razvijena su su dvainženjerska pravca delovanja, tj. dve zasebne oblastirada. Prva se odnosi na modliranje seizmičkih dejstavasa ciljem uzimanja u obzir njihove prostorne promen-

ljivosti. Provereno je i dokazano da uključenje rotacionihkomponenti može da utiče na efekte od dejstava i da

rezultati dovedu do nesigurnih projektnih rešenja.Procena je izvršena korišćenjem metode spektraodgovora. U drugoj oblasti izučava se upotreba metodeprojektovanja na bazi seizmičkih performansi kaosredstva provere i potvrde za seizmičkim pojačanjimakao i potvrde njihove efikasnosti. Da bi se ovo uradilo,upoređeni su spektar ciljanog kapaciteta i stvarni spektarkapaciteta. Slučaj nedovoljne elastične krutosti inedovoljne čvrstoće pri tečenju smatra se najtipičnijimseizmičkim slučajem kada je potrebno izvestipojačavanja. Diskutovane su obe ove teme i formulisanisu zaključci koji se odnose na filozofiju Evropskihstandarda.

Ključne reči: Projektovanje seizmički otpornihkonstrukcija prema Evrokodu 8 (EN 1998), uzimanje uobzir prostornu promenljivost seizmičkih dejstava,seizmičko projektovanje zasnovano na performansama






20

design seismic force (see above)

( )ik k ik i e i E m hS T θ θ = Φ Γ (A-10)

modal participation factor, mode i

[ ] *

T

i

i

i

m

m

θ

θ ν Φ

Γ = (A-11)

total mass, rotational motion

[ ] ( )2T

Tot k k

k

M m mθ θ θ θ ν ν ν = = ∑ (A-12)

APPENDIX B

APPLICATION OF RESPONSE SPECTRUM METHODTO PEAK RESPONSE EVALUATION OF

EQUIVALENT SINGLE DEGREE OF FREEDOMSYSTEM

The general design strategy used in the paper isbased on mixed application of Capacity SpectrumMethod (CSM) [6], [9], [17] and N2 method [11], [13],[16]. A mixed strategy to determine seismic demandsassuming design seismic action is demonstrated in [5],[13] and [16]. The mixed application of both CSM and N2methods has serious advantage because allows fordetermination of elastic and inelastic target parametersand their demands and the process of displacement iscontrolled through the ductility.

The governing system of equations is presented inthe following usual presentation:

[ ] [ ] g m u f m uν + = − (B-1)

where [ ]m is diagonal matrix containing storey masses

as diagonal elements, ν is the vector of the

transferred horizontal seismic motion (all components

have unit value), g u represents the horizontal ground

acceleration record, u is the vector of horizontal storey

accelerations relative with respect to the base. Thevector f is containing the storey resisting forces which

are dependent on storey displacements.Inelastic performance of the frame structure isstudied by means of plastic mechanism formation usingpushover analysis. Plastic mechanism is obtained onoutput of the pushover analysis and visualized by thevector Φ of dimensionless lateral floor displacements.

The basic assumption for lateral floor displacementsdistribution is written in the form:

( ) ( )u t u t ≈ Φ (B-2)

where ( )u t is the roof displacement as a function of the

time. Equation (B-2) transforms the differential system ofequations representing the behaviour of the dynamicmodel as multi degree of freedom system into a singleequation. It is related to equivalent single degree offreedom (ESDOF) system. This transform can be carriedout only after formation of the plastic mechanism. Thefollowing parameters are calculated for further steps:

mass parameter

[ ] ˆ T

m m= Φ Φ (B-3)

mass of ESDOF system

[ ] 3

*

1

T

i i

i

m m mν =

= Φ = Φ∑ (B-4)

internal force of ESDOF system

3*

1

T

i ii f f f == Φ = Φ∑

(B-5)

modification factor

[ ]

[ ]

3

*

1

32

1

ˆˆ

T i i

i

T

i i

i

mmm

m m m

ν =

=

ΦΦ

Γ = = =Φ Φ Φ

∑

∑ (B-6)

basic ESDOF equation of the motion

( ) *

*ˆ g

u t f u

m+ = −

Γ

(B-7)

displacement of ESDOF system

( ) ( )*

ˆ

u t u t =

Γ (B-8)

peak total horizontal acceleration

*

* ˆ g

maxmax

f uu

m= +

Γ

(B-8)

displacement ductility demand

demand demand

y

u

uµ = (B-9)

displacement demand for ESDOF / actual system

*

ˆdemand

demand

uu =

Γ (B-10)




21

TESTIRANJE FENOMENA PUZANJA MEKE STIJENE

TESTING OF CREEP PHENOMENA ON SOFT ROCK

Zvonko TOMANOVI Ć ORIGINALNI NAUČNI RAD

ORIGINAL SCIENTIFIC PAPERUDK: 624.131.54

1 UVOD

Test puzanja, po definiciji, karakteriše konstantnoopterećenje pod kojim se materijal s vremenomdeformiše. Uobičajeno je da se testovi puzanja sprovodepovećanjem ili smanjenjem opterećenja na uzorak uinkrementima. Nakon svakog inkrementa, zadržava sekonstantno opterećenje u toku odabranog intervala.Opterećivanje ili rasterećenje uzorka do određenognaponskog nivoa i održavanje konstantnog opterećenjau toku ovako dugog intervala, zahtijeva posebne uređajeprilagođene vrsti materijala koji se ispituje, veličini

uzoraka i naponskom stanju (jednoaksijalno, biaksijalnoili triaksijalno) koje se održava na uzorku. Zasprovođenje testa jednoaksijalnog puzanja, odnosno zaodržavanje konstantnog opterećenja – napona nauzorcima, u toku relativno kratke istorije ovog testa,korišćeno je više različitih uređaja baziranih napolugama i okačenom teretu. U poslednje dvije decenije,konstruisani su i servo-uređaji za konvencionalnetriaksijalne testove puzanja na standardnim cilindričnimuzorcima stijene.

Dug period testiranja i zahtjev da se naponsko stanjeodrži konstantnim, uz relativno mala odstupanja u tokudugog perioda koji se nekada mjeri i godinama, veomakomplikuje uređaje koji treba da odgovore tom zadatku.Pored navedenog, potrebno je obezbijediti pouzdanomjerenje relativno malih deformacija – sve od trenutkaopterećivanja, pa do završetka testa. Pritom, treba imatiu vidu da ukupna deformacija meke stijene može bitiuzrokovana promjenom napona, vremenski zavisnimdeformacijama (puzanje), promjenom temperature,promjenom vlage u stijeni, promjenom vlažnosti uvazduhu i slično.

Prof. dr Zvonko Tomanović, Građevinski fakultet,Univerzitet Crne Gore, Cetinjski put bb, 81000 Podgorica,Crna Gora, e-mail: [email protected]

1 INTRODUCTION

Creep test by definition represents a deformation ofmaterial with time under a constant stress. Commonly,creep tests are performed by incrementally increasing ordecreasing the load on the specimen. After eachincrement, the constant load is maintained for a definedtime interval. Application of stress or strain relief ofspecimen and maintaining constant load over this longtime interval requires use of special devices thatcorrespond to the type of tested material, specimen sizeand stress state (uniaxial, biaxial or three-axial)

maintained in the specimen. In its quite short history,various dead weight and lever arm testing devices havebeen used for the purposes of uniaxial creep tests thatare for maintaining constant stress state within thespecimen. In the last two decades servo-hydraulicequipment has been designed for the conventionalthree-axial creep tests of standard cylindrical rockspecimens.

The long time interval needed as well as therequirement for maintaining the constant stress statewith relatively small variations over the long time intervalwhich, sometimes can be measured by years,significantly complicate the devices that should responseto this task. Additionally, it is necessary to enablereliable measuring of relatively small deformations fromthe moment the load is applied up to the test completion.Here it should be noted that total deformation of soft rockcan be caused by stress changes, time-dependentdeformations (creep), temperature changes, moisturechanges, air humidity changes etc.

In addition to the above mentioned a change in therock moisture content alone, without any changes in

Prof. dr Zvonko Tomanovic, Faculty of Civil Engineering,University of Montenegro, Cetinjski put bb, 81000Podgorica, Montenegro, e-mail: [email protected]




22

Pored navedenog, samo promjena vlažnosti stijenebez promjene napona ili temperature, može biti uzrokdeformacija puzanja. Ovaj efekat, u literaturi poznat kao„hidraulično omekšanje”, među prvim uočili su Griggs iBlacic ( 1965) i Griggs (1967). Efekte vlažnosti vazduhaoko uzoraka kamene soli pri testu puzanja izučavali suHunsche i Shulze (1996). Ustanovljen je veliki uticaj napuzanje kamene soli, kada se vlažnost vazduha kreće u

granicama od 0% do 65%. Dosad, u literaturi nijedokazano da vlažnost u vazduhu značajno utiče napuzanje drugih vrsta stijena (sistematizovano kod:Cristescu, [4] ). Aparatura i procedure koje se sprovode utoku testiranja, treba da omoguće da se ukupnavremenska deformacija, koja je posljedica prethodnenaponske promjene, razdvoji od ostalih komponentivremenskih deformacija koje se razvijaju u stijeni u tokuvremena (skupljanje zbog gubitka vlage, temperaturnedilatacije itd.).

Puzanje u stijeni počinje značajnije da se proučavapočetkom šezdesetih godina XX stoleća, nakonpionirskih radova Michelson-a (1917) koji je izvodiotestove torzije na stijeni (po uzoru na ispitivanje metala).Početak istraživanja puzanja stijena karakterišu problemiu vezi s tehnikom izvođenja i interpretacijom rezultatatesta. Testove puzanja pri jednoaksijalnoj kompresijiizvodilo je više istraživača, na primer Evans i Woo( 1937), na granitu, mermeru i soli. Nichihara (1958),Hardy (1959, 1966) i Price (1964) istraživali su puzanjesedimentnih stijena (sistematizovano kod: Jaeger et al.,[7] ).

Šezdesetih i sedamdesetih godina prošlog vijeka,značajan broj testova izveden je u BGR-u u Hanoveru, uNjemačkoj (BGR – Bundesansalt fur Geowissenschaftenund Rohstoffe – Savezni zavod za geonauke i sirovine),na kamenoj soli pri jednoaksijalnom i triaksijalnom stanjunapona. Gimm (1968), Dreyer (1974) i Baar (1977) izvelisu većinu testova puzanja stijenske mase „in situ” u

rudnicima kamene soli. Na osnovu rezultata mjerenjakonvergencije u hodnicima rudnika kamene soli, reološkizakoni ponašanja kamene soli bili su predmet izučavanjaHoeffer-a (1958), Schuppe-a (1961), Potts-a i Hendley-a(1965) – sistematizovano kod: Langer, [8] . Laporci ilaporovite stijene se u literaturi, kao predmet izučavanjavremenski zavisnih deformacija – puzanja, pojavljujusporadično. Cristescu prikazuje rezultate jednoaksijalnogtesta puzanja na glinovitom laporcu u trajanju od 7–8dana [4] . Bergues i Tomanovi ć izučavali su puzanjetvrdih laporaca [1], [14] i [15].

Istraživanja vremenski zavisnih deformacija stijena(puzanje) praktičnu primjenu imaju pri analizi različitihnaponsko-deformacijskih fenomena u stijenskoj masi, ali

je najčešća pri analizi stabilnosti i deformacija okotunelskih otvora. Mjerenjem konvergencije radijalnihdeformacija konture tunelskog otvora, s jedne strane,ocjenjuje se stepen uspostavljenog balansa između silau stijenskoj masi i podgradnoj konstrukciji, a s drugestrane – prethodnim analizama procjenjuju se ukupnevremenske deformacije koje će se desiti oko tunelskogotvora ([2], [3], [13], [10], [11], [12]).

stress and temperature, can be a cause of the creepdeformation. This effect, known as hydraulic weakeningin literature, was among first ones identified by Griggsand Blacic (1965) and Griggs (1967). Effects of airhumidity on the rock salt specimens at creep test werestudied by Hunsche and Shulze (1996). A significantimpact on the rock salt creep was identified in casewhen air humidity varied within range of 0-65%. The up

to date literature records no proof of any significantimpact of air humidity on the creep behaviour of otherrocks (systematised at: Cristescu [4]). Apparatus andprocedures used in the course of testing should enableseparation of total time-dependent deformation, whichresults from the stress change from other components of time-dependent deformations which may occur withinthe rock over time (shrinkage due to moisture change,temperature change etc.).

In the early 60s, more consideration was given to thestudying of a rock creep following the pioneering worksof Michelson (1917), who carried out torsion tests of rock(by example of metal testing). The beginnings of the rockcreep studies are marked by problems related to thetesting technique interpretation of test results. Uniaxial

compression creep tests were carried out on granite,marble and salt by many researchers such as Evans andWood (1937). Nichihara (1958), Hardy (1959, 1966) andPrice (1964) studied creep behaviour of sediment rocks(systematised at: J. C. Jaeger et al., [7] ).

In the 60s and 70s of the last century, a greatnumber of uniaxial and three-axial compression tests ofrock salt was carried out at the BGR (BGR -Bundesansalt fur Geowissenschaften und Rohstoffe -Federal Institute for Geosciences and NaturalResources) Gimm (1968), Dreyer (1974) and Baar(1977) carried most of the creep tests of rock mass insitu in the rock salt mines. Based on the results ofconvergence measurements in the corridors of the rock

salt mine, rheological behaviour of soft rock became asubject of research performed by Hoeffer (1958),Schuppe (1961), Potts and Hendley (1965),(systematised at: Langer, [8]). The literature records onlysporadic occurrences of the marl and marlstone in termsof studying of the time-dependent deformations (creep).Cristescu (1993) has presented results of the uniaxialcompression creep test of clayish marl carried out overthe time interval of 7-8 days [4]. Bergues andTomanovichave studied creep behaviour of hard marls[1], [14] and [15].

.Research of time-dependent rock deformations(creep) finds its practical application in analysis ofdifferent strain-dependent deformation phenomena in

rock mass, yet it is mostly applied for the purposes ofanalysis of stability and deformations in the rock aroundthe tunnel opening. Measurement of convergence ofradial deformations of the tunnel opening contourenables to assess the achieved balance degree betweenthe forces in rock mass and support system on one side,while on the other side, preliminary analyses are used toestimate total time-dependent deformations that mayoccur in the rock around the tunnel opening ([2], [3], [13],[10], [11], and [12]).




23

2 URE ĐAJI ZA ODRŽAVANJE KONSTANTNOGNAPONSKOG STANJA PRI TESTOVIMAPUZANJA NA MEKOJ STIJENI

Za test puzanja, odnosno održavanje konstantnogopterećenja, do danas su korišćene dvije dominantnevrste uređaja: ramovi s primjenom poluga sa okačenim(„mrtvim”) teretom i uređaji s hidrauličkim servo-kontrolisanim silama. Prednost uređaja s „mrtvim“teretom jeste to što su jednostavne konstrukcije ipouzdani u toku dugog perioda, a prednost hidrauličkihsistema je što su malih gabarita. Za jedan broj testiranjakorišćeni su i nešto manje pouzdani uređaji koji suzahtijevali manuelnu korekciju sile u toku vremena(dotezanjem matice i/ili opruge).

Poseban problem pri testu puzanja predstavljakonstruisanje ramova za održavanje konstantnognaponskog stanja na modelima, kao što su modelikojima se simulira tunelski iskop pri dvoosnom stanjunapona. Za realizaciju eksperimentalnog laboratorijskogistraživanja puzanja laporca koji se prezentuje u ovomradu, korišćeni su posebno konstruisane nestandardaneaparature i uređaji za održavanje konsatantnog

naponskog stanja (prema nacrtima autora ovog rada).Za pojedine segmente istraživanja korišćena je istandardna laboratorijska oprema kao što su:laboratorijske prese, mjerači sile, deformetri i tako dalje.U nastavku teksta, opisuju se uređaji i instrumentikorišćeni za test puzanja laporca pri jednoaksijalnom ibiaksijalnom stanju napona.

2.1 Uređaj sa okačenim („mrtvim”) teretom za jednoaksijalno opterećivanje

Za održavanje konstantnog opterećenja na jednoaksijalnim prizmatičnim uzorcima laporca dimezija15x15x40 cm korišćen je uređaj s dvostrukom polugom,kapaciteta 750 kN (slika 1 i slika 2). Okačeni teret nakraju poluge 2 prenosi se na polugu 1 i aplicira nauzorak kao sila pritiska. Zavisno od položaja uzorka dužpoluge 1, težina tega multiplicira se od 10 do 150 puta.Za ispitivanje uzoraka laporca poprečnog presjeka15x15 cm i jednoaksijalne čvrstoće oko 8.8 MPa,optimalan položaj uzoraka je pri multipliciranju „mrtvog”opterećenja za oko 100 puta.

Važna faza pri postavljanju testa puzanja jeste icentrisanje opterećenja. Neophodno je da seopterećenje na uzorak nanese što ravnomjernije, tako dapočetna deformacija uzorka indukuje minimalneinklinacije poprečnog presjeka. Postolja sa četiri zavrtnjaomogućavala su da se uzorak podiže naviše, kako bi se

zatvorili svi zazori između pojedinih djelova poluga irama i mjerača sile (vidi sliku 2.b) i obezbijedile približnoiste deformacije na sve četiri strane prizmatičnihuzoraka.

Svaki uređaj od šest uređaja za jednoaksijalnetestove puzanja opremljen je prstenastim mjeračem silekapaciteta 250 KN, posebno konstruisanim iproizvedenim za ovo eksperimentalno istraživanjepuzanja laporca. Opterećenje na uzorak aplicirano jeodozgo preko prstenastog mjerača sile, koji jeistovremeno poslužio i kao zglob (slika 1 i 2.a). Na tajnačin, efekti dodatne ekscentričnosti opterećenja,uzrokovane rotacijom poluge 1 u toku spuštanja „mrtvog”tereta, svedeni su na zanemarljive vrijednosti.

2 DEVICES FOR MAINTAINING CONSTANT STRESSSTATE AT CREEP TESTING OF SOFT ROCK

For the creep test, that is for maintaining theconstant load, two dominant device types have beenused up to date - lever arm system in dead weight creepframe and hydraulic servo control compression testingdevice. The advantage of the dead weight creep testdevice is its simple structure and reliability over a longtime period, while the advantage of hydraulic system liesin its small overall size. For a certain number of testsdevices of somewhat lower level of reliability have beenused, requiring adjustments of compression in thecourse of time (additional nut and/or lever tensioning).

Particular issue at creep test arises with a view to thedesign of frames for maintaining of the constant stressstate in models, such as the model for simulation oftunnel excavation at biaxial stress state. A speciallydesigned non-standard apparatus and devices formaintaining stress state made by the author of thispaper have been used for the purpose of experimentallaboratory testing of the marl creep presented in thispaper. Certain segments of the research involved the

use of standard laboratory equipment as well - labora-tory presses, dynamometers, deformeters etc. The textbelow describes devices and instruments that have beenused for the creep test of marl at uniaxial and biaxialstress state.

2.1 Device with hanged (dead) weight for uniaxialcompression

For maintaining the constant load on uniaxial prisma-tic marl specimens of 15x15x40cm in size, the two-leverarm device has been used, with capacity of 750kN,Figure 1 and Figure 2. Weight hanged at the end of thelever arm 2 is transmitted to lever arm 1 and then ap-plied to the specimen acting as a compression. Depen-ding on the position of the specimen along the lever arm1, the weight is multiplied by 10 to 150 times. For testingof marl specimen of average section 15x15cm anduniaxial compression strength of approximately 8.8 MPa,an optimum position of the specimen is achieved whendead weight is multiplied by approximately 100 times.

An important phase in setting-up the creep test iscentring the weight. It is required to apply the load ontothe specimen as evenly as possible so to enable theinitial deformation of the specimen to induce minimuminclinations of the cross section. Stands with four boltsprovided for lifting the specimen upwards in order toclose all the gaps between certain lever arms and frame

and dynamometer (see Figure 2.b) and enable approxi-mately even deformations at all four sides of prismaticspecimen.

Each of the six used devices for uniaxial com-pression creep tests is equipped with ring dynamometerwith capacity of 250kN which was specially designedand constructed for the purposes of this experimentalresearch of the marl creep behaviour. The load wasapplied to the top of the specimen by means of ringdynamometer which at the same time served as a hinge(Figure 1 and Figure 2.a). In this way, the effects of theadditional eccentricity of the loading, caused by rotationof lever arm 1 during lowering of the dead weight, havebeen reduced to negligible values.




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Slika 1. Dispoziciona šema uređ aja (rama) za jednoaksijalno optereć ivanjeFigure 1. Layout of device (frame) for uniaxial compression

Slika 2. Uređ aji za: a) jednoaksijalno optereć ivanje; b) dvoosno optereć ivanje

Figure 2. Devices for: a) uniaxial compression; b) biaxial compression

2.2 Uređaj sa okačenim („mrtvim”) teretom zabiaksijalno opterećivanje

Za održavanje konstantnog opterećenja na pločastimbiaksijalno opterećenim uzorcima dimenzija 60x60x10cm korišćen je uređaj s dvostrukom polugom, kapaciteta1500 kN (slika 2b i slika 3), istog sistema prenošenjavertikalnih sila, kao i uređaj korišćen za jednoaksijalnoopterećivanje. Trapezasti čelični elementi postavljeni su

2.2 Device with hanged (dead) weight for biaxialcompression

For maintaining the constant load on biaxial platespecimens of 60x60x10cm in size, the two-lever armdevice has been used, with capacity of 1500kN, Figure2b and Figure 3, with the same system for transmissionof vertical load as used in case of the device for uniaxialcompression. Trapeze-shaped steel elements were placed

a) b)








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Aksijalna sila i bočni pritisak manuelno sukontrolisani dodavanjem tegova, a na osnovu mjerenjasile i pritiska primjenom ćelije za mjerenje sile i ćelije zamjerenje pritiska uz očitavanje na „mjernom mostu”.Primijenjena aparatura pokazala je lako održavanjekonstantnog bočnog pritiska u toku testa i jednostavnorukovanje pri testu s kontrolisanim silama. Takođe,aparatura omogućava jednostavno izvođenje testova

puzanja u konvencionalnoj triaksijalnojćeliji.

by adding weight and based on measurements of theload and strain by means of the compression measuringcell and reading of results on the "measuring bridge".The applied apparatus proved to provide the easymaintaining of the constant lateral strain during the testand that it is simple to operate in case of the strain-controlled test. Moreover, the apparatus enables simpleperformance of the creep tests in the conventional three-

axial cell.

Slika 5. Aparatura za triaksijalni test

Figure 5. Three-axial compression test apparatus

3 MJERENJE SILE I DEFORMACIJE

Instrumenti i uređaji za test puzanja moraju dazadovolje dva osnovna uslova: a) održavanjekonstantnog opterećenja u toku izvođenja testa; b)instrumenti za mjerenje deformacija treba da imajumogućnost mjerenja deformacije od početka testa donekog vremena t (nekoliko dana, mjeseci ili godina).

3.1 Mjerenje sile

Mjerenje sile pri nanošenju opterećenja na uzorke prikratkotrajnim testiranjima (ili testovima puzanja u trajanjuod nekoliko sedmica) i mjerenje sile za testove puzanjakoji traju nekoliko mjeseci ili godina zahtijevaju potpunorazličitu mjernu tehniku. Tako je pri kratkotrajnimtestovima moguće koristiti standardne električne mjeračesile. Međutim, pri dugotrajnim testovima puzanja,mehanički mjerači sile imaju značajne prednosti upoređenju sa električnim, u pogledu kompleksnostiopreme, sistema za održavanje stalnog električnognapona, opreme za akviziciju podataka i cijene.

3 MEASUREMENT OF FORCE AND DEFORMATION

Instruments and devices for creep test should meettwo basic requirements: a) maintaining constant loadwhile the creep test is carried out; b) instruments fordeformation measurement should provide deformationmeasurements from the commencement of the test totime t (several days, months or years).

3.1 Force measurement

Force measurement of load application onspecimens at short-time tests (or creep tests with theduration of several weeks) and measurement of force forcreep tests the duration of which is several months oryears require completely different measurementtechnique. For the short-time tests standard electronicdynamometers can be used. However in the cease oflong-time creep tests, the mechanical dynamometers arefar more suitable compared to the electronicdynamometers in terms of the complexity of equipment,system for maintaining constant electrical tension, dataacquisition equipment and costs.




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Za mjerenje sile koja se aplicira na uzorak i njenukontrolu u toku testa puzanja, pri eksperimentalnomistraživanju puzanja laporca, konstruisani su prstenastimjerači sile (prema nacrtima autora ovog rada). Mjerač sile sastoji se od čeličnog prstena, komparatera tačnosti1/1000 mm i postolja (kako je prikazano na slici 6).Mjerači sile kapaciteta 250 kN korišćeni su za mjerenjesile jednoaksijalno opterećenih prizmatičnih uzoraka, a

kapaciteta 350 kN korišćeni su za mjerenje sila kodpločastih biaksijalno opterećenih uzoraka. Različit

kapacitet mjerača sile posignut je povećanjem širineprstena, koja je iznosila 60 ili 80 mm.

Svaki mjerač sile testiran je u laboratorijskoj presi idefinisan je koeficijent proporcionalnosti između ugibaprstena i na prsten aplicirane sile. Veza ugib–sila jestelinearna: F =k *u, gdje je: F – vertikalna sila, u – vertikalniugib prstena. Koeficijent proporcionalnosti k određen jemetodom najmanjih kvadrata za svaki proizvedenimejerač sile i kretao se u granicama od 462.00 do491.04.

Test puzanja trajao je više od jedne godine, uzvarijaciju temperature od 10oC do 32oC. Radi toga,izvršena je provjera uticaja promjene temperature na

tačnost instrumenta. U opsegu temperature od 10oC do40oC, ustanovljena je greška uzrokovana temperaturnimpromjenama manja od 0.2%, a ukupna greška mjerenjau toku jedne godine je ispod prihvatljive granice od 1%za ovu vrstu testa [4]. Mehanički instrument obezbijedio je mjerenje sile u toku nanošenja opterećenja uinkrementima, ali i kontrolu sile u toku testa puzanja, utrajanju duže od jedne godine, na vrlo jednostavan ipouzdan način. Nakon završetka testa, kazaljkakomparatera, ugrađenog u mjerač sile, vratila se na nulu – što je siguran pokazatelj da je mjerni prsten u tokučitavog testa bio elastično opterećen.

For measurement of the force that is applied to thespecimen and control of such force during the creep test,in the course of experimental research of the marl creepbehaviour, the ring dynamometers have been designedby the author of this paper. The dynamometer consistsof a steel ring, comparator with the accuracy of1/1000mm and stand pedestal as illustrated in Figure 6.Dynamometers with the capacity of 250KN were used

for force measurement in case of uniaxially compressedprismatic specimens, while dynamometers with thecapacity of 350KN were used for force measurement incase of biaxially compressed specimens. Differentcapacities of dynamometer have been achieved byincrease in width of the ring which was 60 or 80 mm.

Each dynamometer was tested in the laboratorypress with the defining of the coefficient of proportionalitybetween the ring deflection and on the ring of the appliedforce. The deflection-force relation is linear: F=k*u,where: F - vertical force, u-vertical deflection of the ring.The coefficient of proportionality k was determined bythe least squares method for each constructeddynamometer and its values ranged between 462.00and 491.04.

The creep test lasted for a period of over one yearwith temperature variations from 10 oC to 32oC.Therefore, the accuracy of the instrument was testedagainst the effect of temperature variations. For thetemperature range from 10oC to 40oC, a measurementerror of 0.2% has been identified due to the changes intemperature, while total measurement error for one yearperiod was beyond the acceptable limit of 1% for thistype of test [4]. A mechanical instrument enabled verysimple and reliable force measurement during theincremental application of load, and also the control offorce during the creep test for a period of over one year.Upon test completion, the hand of the comparatorintegrated in the dynamometer returned to zero position,

which indicated with certainty that the measuring ringremained within the range of elastic stress.

Slika 6. Prstenasti mjerač sile kapaciteta 250 kNFigure 6. Ring dynamometer of capacity 250kN




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3.2 Mjerenje deformacija

Mjerenja dilatacija izvedena su mehaničkimdeformetrom tipa „Pfender” (slika 7 – očitavanjapomjeranja 1/1000 mm na mjernoj bazi 100 mm) ipokazala su bolje rezultate pri testu puzanja, upoređenju s mjernim trakama, u pogledu tačnostimjerenja u toku dugog intervala. Nakon dugotrajnihtestiranja, ustanovljena je greška mjerenja ovog tipauređaja 2-3/1000 mm. Mjerena promjena rastojanjaizmeđu repera u toku testa puzanja na uzorcima odlaporca kretala su se od 0.5 do 1 mm. Imajući u viduostvarene dilatacije, greška mjerenja od 3/1000 mmmože se ocijeniti kao prihvatljiva.

3.2 Deformation measurement

Measurements of dilatations that were performedusing mechanical deformeter type Pfender, Figure 7(reading displacements 1/1000mm on measuring baseof 100mm) showed better results at creep test,compared to strain gauge with a view to themeasurement accuracy over a long time interval. Afterlong lasting tests, the measurement error of 2-3/1000mmwas identified for this type of device. Measured changesin distance between the reference points during thecreep testing of marl varied from 0.5 to 1mm. Taking intoconsideration the occurred dilatations, measurementerror of 3/1000mm can be considered as acceptable

Slika 7. Mjerenje deformetrom tipa „Pfender” na ploč astom uzorkuFigure 7. Measurement of deformations in plate specimen by deformeter type "Pfender"

4 TEHNIKA FIKSIRANJA MJERNIH TAČAKA IMJERENJE DEFORMACIJA

Prizmatični i plo

časti uzorci laporca, glavnih serijamjerenja, opremani su mrežom mjernih mjesta za

mjerenje promjene dužine između mjernih bazamehaničkim deformetrom tipa „Pfender”. Mjerenje ovimdeformetrom zahtijeva lijepljenje mjernih tačaka(mesingana pločica sa utisnutom čeličnom kuglicom)direktno na uzorak. Pritom, mek stijenski materijal iprirodna vlažnost uzorka pričinjavaju dosta poteškoća.Nakon proba učinjenih s nekoliko vrsta ljepkova i načinapripreme podloge na uzorku, ustanovljeno je dafiksiranje mesinganih pločica – mjernog mjesta ljepilomtipa „Hotinger-X60” i lagano hrapavljenje podloge tj.površine uzorka, daju najbolje rezultate. Ovako fiksiranamesingana pločica ostaje adekvatno fiksirana za uzoraku toku perioda mjerenja od godinu dana i pri više

desetina serija mjerenja.

4.1 Prizmatični uzorci

Svi prizmatični uzorci laporca dimenzija 15x15x40cm opremljeni su mjernim tačkama koje su omogućilemjerenje deformacija u pravcu podužne ose uzorka ideformacija u porečnom pravcu (prikazano na slici 8).Mjerni opseg mehaničkog deformetra tipa „Pfender” jeste 1 mm, a odabrana mjerna baza instrumenta iznosioko 100.4 mm. Time je bilo moguće izmjeriti dilataciju od

4 TECHNIQUE OF AFFIXING THE MEASURINGPOINTS AND DEFORMATION MEASUREMENT

Prismatic and plate marl specimens of the mainmeasurement series were provided with the network ofmeasuring points for measuring the change in lengthbetween measuring bases by mechanical deformetertype “Pfender”. Measuring with this deformeter requiresthe measuring points (brass plate with inlaid steel ball) tobe affixed directly to the specimen. Soft rock materialand natural moisture of the specimen caused lot ofdifficulties hereto. Upon trial attempts using severaltypes of adhesives and methods of base preparation onthe specimen, it was observed that fixing of brass plates- measuring points by means of adhesive type “Hotinger-X60” and slight coarsening of the base i.e. surface of thespecimen provided the best results. The brass platefixed in this way remained properly fixed to the specimen

for a measurement period of one year even in the caseof several tenths of measurement series.

4.1 Prismatic specimens

All prismatic specimens of 15x15x40cm in size wereprovided with measuring points, which enabledmeasurement of deformations in the direction of thelongitudinal axis of the specimen and deformations in thehorizontal direction as illustrated in Figure 8. Measuringrange of the mechanical deformeter type “Pfender” is1mm, and selected measuring base of the instrument is




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8 do 9 promila. Pripremljeno je osamnaest prizmatičnihuzoraka – šest uzoraka probne serije i dvanaest uzorakaglavne serije, od kojih je na šest sproveden test puzanjau trajanju oko godinu dana, a ostali su testirani nakratkotrajno opterećenje do loma [15].

about 100.4mm. This enabled measurement of a dilata-tion of 8-9 per mils. Eighteen specimens were prepared:6 specimens of trial series and 12 specimens of mainseries, 6 of out of which were subject to creep testing fora period of about one year, while the remainingspecimens were tested against short-time loading untilfailure occurred [15].

Slika 8. Prizmati č ni uzorak, šema mjernih mjestaFigure 8. Prismatic sample, layout of measuring points

4.2 Pločasti uzorci

Svi pločasti uzorci opremljeni su sa po dvije mrežemjernih tačaka, sa obje strane uzorka, koje su omogućilemjerenje polja deformacija u ravni ploče. Mreža bazemjerenja 100 mm postavljena je na čitavoj površiniuzorka, a mreža baze mjerenja 60 mm formirana je ucentralnom dijelu uzorka (prikazano na slici 9). Gušćommrežom mjernih mjesta obuhvaćen je uži dio oko otvorakoji će se izbušiti u drugoj fazi testa. Mjerenjedeformacije po stranama trouglova mreže omogućujedefinisanje deformacije u bilo kom željenom pravcu.

Postavljanje mjernih baza za mrežu trouglovapredstavlja izuzetno komplikovan zadatak zbog malogmjernog opsega instrumenta (1mm). Da bi se moglaobavljati mjerenja, sve tačke su međusobno morale biti

postavljene na rastojanju 100 mm ±0.2 mm. Nakonpromjene naponskog stanja, mjerene su deformacijepuzanja na mreži mjernih tačaka (u sve tri faze testa) i tonakon 1, 6 i 24 sata od nanošenja opterećenja, zatimnakon tri dana i sedam dana, a u daljem periodu –svakih petnaest dana. Intervali mjerenja odabrani sutako da razlike prethodne i tekuće mjerene deformacijebudu približno jednake.

4.2 Plate specimens

Every plate specimen was provided with twonetworks of measuring points on both sides of thespecimen, which enabled measuring of deformation fieldin the plane of the plate. Network of the measuring baseof 100mm was placed over the total surface of thespecimen, while the network of the measuring base of60mm was formed in the central part of the specimen, asillustrated in Figure 9. A network with higher density ofmeasuring points covered the immediate area aroundthe opening which would be bored in the phase 2 of thetest. Measurement of deformation on the sides oftriangles of the network enables the deformation to bedetermined in any preferable direction required.

Setting the measuring bases for the network of

triangles is a quite complicated task due to the smallmeasuring range of the instrument, this being 1mm. Itwas required for all points to be set at the mutualdistance of 100mm ±0.2mm for measurement purposes.Upon the change of stress state, creep deformationswere measured on the network of measuring points a (inall three phases of the test), within 1, 6 and 24 hoursafter application of the load, then after 3 and 7 days andon every 15 days in the further period. Measuringintervals were selected to ensure approximately evendifferences in previously and currently measureddeformation.




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Slika 9. Ploč asti uzorak: a) šema mjernih mjesta; b) uzorak za kompenzaciona mjerenjaFigure 9. Plate specimen: a) layout of measuring points; b) specimen for compensation measurements

4.3 Triaksijalni uzorci

Za mjerenje deformacija uzoraka u konvencionalnomtriaksijalnom testu, u standardnoj Hook-ovoj ćeliji,upotrijebljene su mjerne trake i rozete koje mjeredeformacije u dva upravna pravca (proizvođač: TML-Japan). Za lijepljenje rozeta korišćeno je ljepilo P2 (TML-Japan), koje ima veliku rastegljivost, što omogućavamjerenje velikih dilatacija kakve su očekivane napredmetnom materijalu.

Svaki uzorak opremljen je sa po tri rozete ili trake,postavljene u sredini visine uzorka, pod centralnimuglom 120o, posmatrano u poprečnom presjeku uzorka(kako je prikazano na slici 10). Za povezivanje traka s„data loger-om” korišćen je trakasti kabl debljine oko 0.3mm. Nakon postavljanja kabla i povezivanja s rozetama,

preko uzorka je navlačena zaštitna membrana.Pripremljena su i ispitana ukupno dvadeset četiri uzorkau triaksijalnom aparatu, uz mjerenje deformacija, a jošdesetak uzoraka dovedeno je do loma bez mjerenjadeformacija (uz mjerenje aksijalne sile loma i bočnogpritiska) [16].

4.3 Tree-axial specimens

Measurement of the specimens’ deformations byconventional tree-axial test in the standard Hook's cellincluded the use of a strain gauge and rosettes fordeformation measurement in the two normal directions(accuracy 10-3mm, producer TML - Japan). Glue (TML -Japan), with a high deformation power, was used for thegluing of the rosettes, which enabled the measurementof the high strain expected in the examined material.

Each specimen was equipped with tree rosettes or astrain gauge set in the middle of specimen height underthe central angle of 120o, seen through the cross-section of the specimen, as shown in Figure 10. A stripecable, about 0.3mm thick, was used to connect the straingauge with the "data logger". After the cable was set and

the rosettes were connected, a protective membranewas put over the specimens. A total of 24 specimenswere prepared and examined in the tree-axial devicealong with deformation measurement and about tenmore specimens were led to failure without anydeformation measurement (with a measurement of theaxial stress of the failure and lateral pressure) [16].

Slika 10. Cilindri č ni uzorak za konvencionalni triaksijalni test;a) dispozicija mjernih rozeta; b) rozeta postavljena na uzorak

Figure 10. Cylindrical specimen for a conventio0nal tree-axial test;a) disposition pf measurement rosetta; b) rosettes attached to the specimen

Mjerne trakeStain guage




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5 NEOPTEREĆENI UZORCI ZA MJERENJEDEFORMACIJE SKUPLJANJA I DEFORMACIJEOD PROMJENE TEMPERATURE

5.1 Osnovne fizičko-mehaničke osobine ispitivanoglaporca, mineralni i hemijski sastav

Testovi puzanja sprovedeni su na sivom cementnomlaporcu. Prosječna vrijednost suve zapreminske težineovog laporca, iz kojega su uzorci uzeti, iznosi oko 18.8kN/m3 (od 17.2 do 20.38 kN/m3), a prosječna pritisna jednoaksijalna čvrstoća monolita oko 8.8 MPa. Ispitivanilaporac – u pogledu mineralnog sastava – dominantnopredstavljaju kalcit (46–48%) i kvarc (12–13%), dok su uokviru glinovite faze – ilit+smektit, montmorijonit, kaolinit,glaukonit, transformisani feldspat i liskun. Premahemijskom sastavu, sadržaj CaCO3 je u granicama48.10–48.30%, dok je sadržaj nerastvornog ostatka(glinovito+kvarc) u granicama 51.03–51.87%.

5.2 Promjene vlažnosti u toku dugotrajnog

testiranjaNakon vađenja iz prirodno vlažne sredine, na sobnoj

temperaturi, laporac brzo gubi vlagu. Zbog znatnogsadržaja glinovitih minerala i velike poroznosti, posledicagubitka vlage u uzorcima jeste skupljanje materijala. Priintenzivnom sušenju i većem stepenu gubitka vlage,skupljanje uzrokuje pojavu pukotina i raspadanjematerijala. Deformacije koje se javljaju kao posljedicagubitka vlage tj. skupljanja, imajući u vidu ciljistraživanja, bilo je neophodno svesti na prihvatljiv nivo,kako ne bi imale uticaj na mehaničke karakteristikematerijala. Radi redukovanja promjene vlažnosti u tokutesta puzanja, nakon postavljanja mjernih baza (pločica),na uzorak je nanošen tanak sloj parafina. Promjena

vlage ovom mjerom je znatno redukovana, pa jeskupljanje stijenskog materijala, u toku ispitivanja odgodinu dana, svedeno na prihvatljiv nivo – od nekolikoprocenata.

U toku testova puzanja, pri svakoj seriji mjerenjadeformacija, mjereni su temperatura i vlažnost vazduhau laboratoriji u kojoj su rađeni testovi. Vlažnost uzorkaodređivana je metodom sušenja na temperaturi od105oC. Prirodna vlažnost uzoraka kretala se od 11 do12%. Nakon završetka testa puzanja probne serijeprizmatičnih uzoraka, koji je trajao tri mjeseca,ustanovljena je prosječna vlažnost od 10.65%, a nakonzavršetka testa puzanja glavne serije mjerenja, koji jetrajao dvanaest mjeseci, izmjerena je prosječna vlažnost

7.40%.

5.3 Uzorci za mjerenje deformacije skupljanja ideformacije od promjene temperature

Pored preduzetih mjera da se spriječi promjenavlage u uzorcima za vrijeme dugotrajnih testova puzanja,dolazi do izvjesne promjene vlage u uzorcima, a time ido deformacije skupljanja uzrokovane gubitkom vlage.Takođe, nije bilo moguće obezbijediti konstantnutemperaturu u laboratoriji u toku godinu dana, te je

5 NON-LOADED SAMPLES FOR MEASUREMENTOF SHRINKAGE DEFORMATION AND DEFOR-MATION DUE TO TEMPERATURE CHANGE

5.1 Basic physical and mechanical properties ofthe tested marl, mineral and chemicalcomposition

Creep tests were carried out on gray cement marl. Average value of dry volumetric weight of this marl,which was sampled, is approximately 18.8kN/m3 (from17.2-20.38 kN/m3), and average uniaxial compressionstrength of monoliths is about 8.8MPa. In terms to themineral composition, the marl is represented by calcite(46-48%) and quartz (12-13%), and within the scope ofthe clayish phase - illite-smectite, montmorillonite,kaolinite, glauconite, transformed feldspar and mica.With a view to its chemical composition, CaCO3 is withinthe limits of 48.10 - 48.30%, while the remaininginsolvable content (clayish + quartz) is within the limits of 51.03 - 51.87%.

5.2 Change in moisture during long-time testingHaving been extracted from the natural moist

environment, marl loses moisture quickly at roomtemperature. Due to a significant content of clayishminerals and high porosity, the loss of moisture in thespecimens results in shrinkage of material. At intensivedrying and high moisture loss conditions, the shrinkagecauses cracking and decomposition of material. With aview of the objective of the research, it was necessary toreduce deformations resulting from the moisture loss, i.e.shrinkage to the acceptable level to prevent their impacton mechanical properties of the material. For reductionof moisture changes during the creep test, after placingthe measuring bases (plates) a thin layer of paraffin was

applied to the specimen. This measure enabled signi-ficant reduction in moisture change, hence the shrinkageof rock material during the one year period of testing,was reduced to the acceptable level of several percents.

In the course of creep tests, at every deformationmeasuring series, temperature and air humidity weremeasured in the laboratory in which the tests werecarried out. Humidity of specimen was defined by amethod of drying at temperature of 105oC. Naturalhumidity of specimen varied in the range from 11% to12%. Upon completion of the creep test of trial series ofprismatic specimens, which lasted three months, themeasured average humidity was 10.65%. After thecompletion of creep test main series of measurements,

which lasted for 12 months, the measured averagehumidity was 7.40%.

5.3 Specimens for measurement of shrinkagedeformation and deformation due totemperature changes

Despite the measures taken to prevent changes inmoisture within samples during the long-time creeptests, certain moisture changes in the samples stillappear, and consequently the shrinkage deformationscaused by the moisture loss occur. Moreover, it was






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6 REZULTATI TESTIRANJA

6.1 Test jednoaksijalne kompresije laporca prikratkotrajnom opterećenju

Jednoaksijalni test na prizmatičnim uzorcima, prinespriječenom bočnom širenju, kao najjednostavniji zaizvođenje i interpretaciju, obično predstavlja početnikorak pri istraživanju naponsko-deformacijskogponašanja čvrstih materijala. Iako s vrlo jednostavnimpoljem napona i deformacija, ovaj test daje vrlo jasnematerijalne konstante i/ili parametre koji se mogu korisnoupotrijebiti i pri analizi veoma složenih naponski zavisnihfenomena.

Ispitivani uzorci laporca, pri jednoaksijalnom testu,pokazuju nelineranu vezu između napona i deformacijaovog materijala, čak i pri malom nivou napona. Na slici12 prikazan je reprezentativan ε σ − dijagram laporca,dobijen pri jednoaksijalnom testu. Uzorci su opterećivani jednoaksijalno, u pravcu duže ose prizme, uinkrementima od 1.0 MPa. Deformacije su mjerene upodužnom i poprečnom pravcu. Pri svakom inkrementusile, deformacije su mjerene jedan minut nakon

nanošenja opterećenja. Brzina opterećivanja je oko 3–4minuta po inkrementu, tako da je uzorak dovođen doloma za oko 30 minuta.

Brojne stijene pokazuju osobinu da imaju različitadeformabilna svojstva u različitim pravcima ispitivanja.Ova osobina je prvenstveno posljedica teksture tj.slojevitosti ili orijentacije zrna u mikrokompozicijistijenskog materijala. Generalno je poznato da stijenskimaterijal pokazuje veću deformabilnost u pravcuupravnom na slojeve, a manju u pravcima paralelnimravnima slojevitosti. Ispitivanje naponsko deformacijskeveze laporca u različitim pravcima izvedeno je naprizmatičnim uzorcima pri jednoaksijalnom testu snespriječenim bočnim širenjem. Uzorci su rezani u tri

ortogonalna pravca: jedan vertikalni pravac kojiodgovara vertikalnoj osi neporemećene stijene „in situ” iu dva upravna horizontalna pravca.

Na dijagramu slike 12, prikazani su rezultati jedno-aksijalnih testova (oznaka EZ ukazuje na opterećivanjeupravno na slojeve, EX u ravni slojeva; oznaka v i hukazuje na pravac mjerenja dilatacija; v - u pravcu dej-stva opterećenja, a h - u upravnom pravcu). Jasno seuočava veća deformabilnost laporca u pravcu upravnomna slojeve u odnosu na druga dva ortogonalna pravca.Modul elastičnosti (sračunat pri nivou napona 2.0 MPa)u pravcu X dva puta je veći od modula elastičnosti u

pravcu Z, 0.2/ = x z E E . Početni moduli elastičnosti u x-y

ravni, pokazuju jednake vrijednosti (horitontalna ravan

„in situ”). Navedeno deformacijsko ponašanje laporcaukazuje na to da se adekvatni proračuni naponskizavisnih fenomena u laporcima moraju sprovoditireološkim modelima koji obuhvataju efekte transverzalneizotropije.

6 TEST RESULTS

6.1 Uniaxial compression test of marl at short-termloading

Uniaxial compression test of prismatic samples, withunconstrained lateral expansion, being the simplest tocarry out and interpret usually represents the initial stepin research of strain-dependent deformation behaviourof rigid materials. Notwithstanding its quite simple field ofstress and deformations, this test provides very clearmaterial constants and/or parameters which can have apurposeful application even for an analysis of highlycomplex strain- dependent phenomena.

Tested marl specimens, at uniaxial compression test,show non-linear relation between the stress anddeformation of this material even at low stress level.

Figure 12 illustrates a typical ε σ − diagram of marlobtained from the uniaxial compression test. Uniaxialload was applied to the specimens in the direction of thelonger axis of the prism, in increments of 1.0MPa.Deformations were measured in the longitudinal andtransverse direction. At each increment, the deformationforces were measured one minute after the loadapplication. With the loading speed of 3-4 minutes perincrement, every specimen was brought to failure withinapproximately 30 minutes.

The feature common in large number of rocks showsdifferent deformability properties in different directions oftesting. This feature comes as a result of the textureand/or layering or orientation of grain in micro-composition of rock material. It is generally known thatthe rock material shows higher deformability in thedirection which is perpendicular to the layers, and lowerdeformability in the directions that are parallel to thelayering planes. Testing of strain-dependent deformationrelation of marls in different directions was carried out on

prismatic specimens at uniaxial compression test withconstrained lateral expansion. The specimens were cutin three orthogonal directions: one vertical directionwhich corresponds to the vertical axis of the undisturbedrock in situ and two perpendicular horizontal directions.

The Diagram illustrated in Figure 12 shows theresults of uniaxial compression tests (where EZ meansloading perpendicular to the layers; EX means loading inthe layering plane; v and h denote direction of thedilatation measurement; v – in the load effect direction; h- in the perpendicular direction). It is easily observed thatthe marl exhibits higher deformability in the directionperpendicular to the layers compared to the other twoorthogonal directions. Elastic modulus (calculated at the

stress level of 2MPa) in the X direction is twice largerthan the elastic modulus in the Z direction, 0.2/ = x z E E .

The initial elastic modules in the x-y plane show evenvalues (horizontal plane in situ). The said deformationbehaviour of marl indicates that proper calculations ofstrain-dependent phenomena in marls need to becarried out on rheological models which include theeffects of transverse isotropy.




35

Slika 12. Rezultati jednoaksijalnih testova upravno na slojeve i u ravni slojevitostiFigure 12. Results of uniaxial compression tests carried out perpendicularly to the layers and in the layering plane

6.2 Rezultati jednoaksijalnih testova pri testupuzanja

Jednoaksijalni test puzanja na laporcu, predstavljenu ovom istraživanju, izveden je na dvije grupe po triprizmatična uzorka dimezija 15x15x40 cm. Ukupnotrajanje testa puzanja bilo je oko 180 dana.Opterećivanje uzoraka izvedeno je u inkrementima od25% od krajnjeg napona testa puzanja, za svaku grupuuzoraka.

Srednje vrijednosti mjerenja deformacija na po triprizmatična uzorka svake različito opterećene grupeuzoraka, pri aksijalnom naponu od 2.0MPa i 4.0MPa,prikazane su na slici 13. Na dijagramu se jasno izdvajazona intenzivnog puzanja materijala u aksijalnom pravcuu prvih dvadeset dana nakon opterećivanja. Priraštaj

deformacija u ovom periodu jeste nelinearan u odnosuna vrijeme. Nakon ovog perioda, deformacije puzanja sumanje i priraštaj deformacija je približno linearan.

Na uporednom dijagramu prikazanom na slici 13, vidise da je – pri aksijalnom naponu od 2.0 i 4.0 MPa –gradijent prirasta deformacije u toku vremena veći koduzoraka opterećenih većim naponom pritiska. Značajno je naglasiti i to da aksijalna deformacija puzanja koja serazvije u toku šest mjeseci dostiže red veličine trenutnedeformacije koju indukuje inicijalna promjena napona.Tako, pri naponu od 2.0 MPa, prosječna trenutnadeformacija (prosječna deformacija na tri uzorka jedneserije) iznosi 1.41o/oo, a prosječna vremenskadeformacija 0.98 o/oo, dok pri naponu 4.0 MPa prosječnatrenutna deformacija iznosi 3.27o/oo, a prosječna

vremenska deformacija 2.74 o/oo.Poprečne dilatacije su oko deset puta manje od

aksijalnih (ako se upoređuju njihove apsolutnevrijednosti), odnosno Poison-ov koeficijent ima vrijednostν = 0.1. Ovaj odnos je približno isti za trenutnu dilatacijuuzrokovanu naponskom promjenom i vremenski zavisnekomponente ukupnih dilatacija. S povećanjem napon-skog nivoa, ovaj odnos raste za ukupnu deformaciju(trenutna + vremenska) u sličnom odnosu kao i kodkratkotrajno opterećenih uzoraka (slika 13). Pri lomu,kod ispitivanog laporca, Poison-ov koeficijent imavrijednost ν = 0.25.

6.2 Results of uniaxial compression tests at creeptest

Uniaxial marl creep test in experimental researchpresented here was conducted on two groups with threeprismatic specimens, 15x15x40cm each. Total durationof the creep test was about 180 days. Loading of thespecimen was conducted in increments of 25% ofdefined level of stress in the creep test, for every groupof specimens.

Average values of deformation measurements onthree prismatic specimens out of each group of speci-mens that were subjected to different loading, at axialcompression load of 2.0MPa and 4.0MPa are illustratedin Figure 13. A zone of an intensive creep of material inthe axial direction in the first twenty days after the load

application is clearly seen in the diagram. Increase indeformations over this time interval is not-linear to time. After this period, the creep deformations are smaller andthe increase in deformations is almost linear.

Comparative diagram illustrated in Figure 13 showsthat at axial compression of 2.0MPa and 4.0MPa, thegradient of deformation increase in time is larger in caseof specimens that are subject to a larger compressionload. It is important to note that axial creep deformationwhich occurs within 6 months reaches the order ofmagnitude of the instantaneous deformation induced byinitial change in compression load. Therefore, at thecompression load of 2.0MPa, an average instantaneousdeformation (average deformation on three samples ofone series) is 1.41 o/oo, and an average time-dependent

deformation is 0.98 o/oo; while at compression load of4.0MPa an average instantaneous deformation is 3.27o/oo,and an average time-dependent deformation is 2.74 o/oo.

Transverse dilatation are about 10 times smallercompared to axial (when absolute values are compared),meaning the value of Poisson ratio is ν = 0.1. This ratiois almost the same in the case of an instantaneousdilatation caused by change in compression and time-dependent component of total dilatations. With theincrease of the compression level, this ratio grows fortotal deformation (instantaneous + time-dependent) inthe same proportion as in the case of short-term loaded




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Slika 13. Dijagram puzanja aksijalno optereć enih prizmati č nih uzorakaFigure 13. Creep diagram of axially compressed prismatic specimens

6.3 Test na pločastim uzorcima

Prilikom testa na pločastim uzorcima (slika 14),nanijeto je opterećenje, jednoaksijalno i biaksijalno usvojoj ravni ploče, u inkrementima od 0.5 MPa, u toku jednog sata, do vertikalnog napona od 2.0 MPa, što jeoko 25% od vršne čvrstoće ispitivanog laporca. Triuzorka opterećena su jednoaksijalno, a u sljedeće trigrupe uzoraka variran je odnos horizontalnog i

vertikalnog pritiska, =vh P P 0.3, 0.5 i 1.0. Inicirano

naponsko stanje (uz održavanje konstantnog odnosa=vh P P const.) održavano je narednih 45 dana, uz

mjerenja polja deformacija na oba lica ploče.

specimens (Figure 13). At the failure of the tested marl,the Poisson ratio is ν = 0.25.

6.3 Test on plate specimens

Test on plate specimens (Figure 14) consisted ofapplying uniaxial and biaxial load on the plate in its ownplane, in increments of 0.5 MPa during a period of onehour up to vertical stress of 2.0 MPa, which makes forabout 25% of the peak strength of the examined marl.Three specimens were loaded uniaxially, while in thefollowing three groups of specimens the ratio

vh P P

between horizontal and vertical load components wasvaried, 0.3, 0.5 and 1.0. The initiated stress state (withmaintained constant ratio between horizontal and

vertical stress) was retained in the following 45 days,with measurement of the deformation fields on bothsides of the plate specimen.

Slika 14. Biaksijalno optereć enje na ploč astim uzorcimaFigure 14. Biaxial load on plate specimens




37

Procedura ove faze testa definisana je tako da seustanovi uticaj bočnog napona na puzanje u vertikalnompravcu, te da se ustanovi varijacija mjerenih deformacija(trenutnih i vremenskih), zavisno od oblika uzorka(prizmatičan/pločast).

Bočno opterećenje biaksijalno opterećenih pločastihuzoraka značajno mijenja deformacijsku sliku, kakopočetne deformacije indukovane naponskom

promjenom, tako i deformacije puzanja. Na slici 15prikazani su rezultati mjerenja deformacije na tri grupeuzoraka opterećenih pri različitom odnosu horizontalnogi vertikalnog napona. Kao uporedna vrednost na svimdijagramima, prikazana je prosječna ukupna vertikalna(PR-v) i horizontalna (PR-h) deformacija prizmatičnihuzoraka pri istom vertikalnom naponu 2.0 MPa.

Pri odnosu 0.1== vhk σ σ kod pločastih uzoraka,

početna i vremenska vertikalna deformacija vrlo malo serazlikuje od deformacije jednoaksijalno opterećenihprizmatičnih uzoraka, kako je prikazano na dijagramuslike 15. Prirast vremenskih deformacija nešto je većikod pločastih uzoraka nego kod prizmatičnih, i to nakon10–15 dana od opterećivanja. Veći prirast vremenskih

deformacija, prema tome, jeste posljedica većeg sekun-darnog puzanja kod pločastih uzoraka (sekundarnopuzanje dobija na značaju s protokom vremena, dokprimarno puzanje, nakon desetak dana, praktično nemadalji uticaj na prirast deformacija). Ove razlike suočigledno posljedica oblika uzorka, jer je ispitivanjeizvršeno pod istim uslovima na istom materijalu.

Sa smanjenjem odnosavhk σ σ = od vrijednosti 1.0

prema jednoaksijalnom stanju, prirast inicijalnih ivremenski zavisnih vertikalnih deformacija raste, kao što je prikazano na dijagramima slike 15, što je posljedicaoslobađanja bočnog širenja uzorka.

The procedure of this phase of the test was designedto determine effects of the lateral stress on the creep inthe vertical direction as well as to determine variations ofthe measured deformations (instantaneous and time-dependent) depending on the shape of specimen i.e.prismatic – plate shaped.

Lateral loading in the case of biaxially compressedplate specimens significantly changes the deformation

behaviour of both the initial deformations induced by astress change and creep deformations. Figure 15 showsresults of deformation measurements on three group ofspecimens loaded at different horizontal-vertical stressratio. A comparative value presented in each diagram isan average total vertical (PR-v) and horizontal (PR-h)deformation of prismatic specimens at the same verticalstress of 2.0MPa.

At ratio 0.1== vhk σ σ in the case of plate speci-

mens, the initial and time-dependent vertical deformationslightly defers from the deformation of uniaxiallycompressed prismatic specimens, as illustrated indiagram, Figure 15. Increment of time-dependentdeformations is somewhat larger in the case of plate

specimens compared to prismatic specimens, after 10-15 days following the load application. Therefore, largerincrement of time-dependent deformations is aconsequence of larger secondary creep of the platespecimens (secondary creep grows in line with time,while primary creep after about 10 days practically hasno further impact on the increase of deformations).These differences obviously come as a result of theshape of specimen, given that the tests were carried outunder the same conditions on the same material.

With decrease of the ratiovhk σ = from the value

of 1.0 towards the uniaxial state, the increment of initialand time-dependent deformations grows, as illustrated inFigure 15, which is a consequence of the unconstrained

lateral expansion of the specimen.

-0,5

0,0

0,5

1,0

1,5

2,0

2,5

3,0

3,5

0,00 5,00 10,00 15,00 20,00 25,00 30,00 35,00 40,00 45,00 50,00

Dilataci

ja/Str ain[‰]

Vrijeme [dani] / Time [days]

PR-v

PR-h

K=0.3 V

K=0.3 H

K=0.5 V

K=0.5 H

K=1.0 V

K=1.0 H

Slika 15. Dijagram puzanja biaksijalno optereć enih uzoraka (PR– označ ava prizmati č an jednoaksijalno optereć enuzorak, a K označ ava odnos horizontalnog i vertikalnog napona na ploč astim uzorcima)

Figure 15. Creep diagram of biaxially loaded specimens (PR means uniaxially loaded prismatic specimen, K means ratioof horizontal and vertical stress on plate specimens)




38

Horizontalna deformacija, kod biaksijalno optere-ćenih pločastih uzoraka, pokazuje kolebanje u prvih 5–7dana od opterećivanja. U kasnijem periodu, puzanje uhorizontalnom pravcu može se evidentirati samo kod jednako vertikalno i horizontalno opterećenih uzoraka.Kod ostalih uzoraka odnosa 5.0== vhk σ σ horizontalne

deformacije u toku vremena zadržavaju vrijednost inici- jalne deformacije, a kod odnosa 3.0== vhk σ σ horizon-

talne deformacije su skoro iste kao i horizontalnedeformacije prizmatičnih uzoraka. Pri odnosu

vhk σ σ = =1.0 i 0.5, horizontalna deformacija jeste

deformacija skupljanja, a pri odnosuvhk σ σ = =0.3

horizontalna deformacija jeste deformacija širenja.

6.4 Konvencionalni test triaksijalne kompresije pridugotrajnom opterećenju

Konvencionalni test triaksijalne kompresije dajemogućnost analize uticaja bočnog napona na

deformisanje. Pritom, bočni pritisak 32 σ σ = (koji sevarira u setu testova) obično ne prelazi vrijednost od50% vršne čvrstoće dobijene u jednoaksijalnom testubez spriječenog bočnog širenja. Generalno, ovaj tip testamoguće je sprovesti uz kontrolisane sile kojim se uzorakopterećuje ili uz kontrolisanje brzine aksijalne deforma-cije. Test puzanja, koji je sproveden na laporcu u neštokraćem vremenu (sedam dana), predstavlja specifičantriaksijalnog testa pri kontrolisanoj sili, koja je u ovomslučaju nakon nanošenja konstantna u toku testa.

Test puzanja na laporcu u konvencionalnomtriaksijalnom aparatu sproveden je na deset uzoraka, prikonstanom opterećenju od 2.0 MPa, u toku tri dana, tepri 4.0 MPa, u toku naredna tri dana. Tri uzorka testirana

su bez bočnog pritiska, kod četiri bočni napon je 1.0MPa, a kod tri 2.0 MPa. Ovaj relativno kratak testpuzanja dao je neke kvalitativne pokazatelje ponašanjaispitivane stijene u toku vremena pri dejstvu jednakogbočnog pritiska.

Na dijagramu na slici 16. prikazani su rezultati testapuzanja triaksijalno opterećenih cilindričnih uzoraka prirazličitim bočnim naponima pritiska. Pune linije ukazujuna aksijalnu (vertikalnu) dilataciju, a isprekidana linija nahorizontalnu (radijalnu) dilataciju. Radijalna deformacijakod svih uzoraka, nakon jednog dana od opterećivanja,zadržava dostignuti nivo, bez obzira na intenzitetradijalnog napona u razmatranom domenu, dok aksijalnadeformacija ukazuje na postojanje uticaja bočnognapona na tok vremenskih deformacija [16].

Horizontal deformation in the case of biaxiallycompressed plate specimens exhibits fluctuations in thefirst 5-7 days after the load application. In the laterperiod, the creep in the horizontal direction can beobserved only in specimens with even vertical andhorizontal loading. In the case of other specimens withratio 5.0== vhk σ σ , horizontal deformations retain the

value of the initial deformation over the time, while at

ratio 3.0== vhk σ σ , horizontal deformations are almostthe same as horizontal deformations of the prismaticspecimens. At ratio

vhk σ σ = =1.0 and 0.5, horizontal

deformation is a shrinkage deformation, and at ratio

vhk σ σ = =0.3, horizontal deformation is an expansion

deformation.

6.4 Conventional test of three-axial compressionunder long term loading

Conventional tests of three-axial compression givethe possibility of analysis of influence of lateral stress on

deformations. Lateral pressure 32 σ σ = (which variesin the set of tests) usually does not exceed value of 50%of peak strength obtained in uniaxial test with free lateraldeformations. Generally speaking, this type of test canbe conducted with controlled forces with load against thespecimen or with control of velocity of axial deformation.Creep test carried out on marl in a shorter period of time(seven days), represents a specific type of three-axialtest with controlled force that is stress state which afterthe application remains constant during the test.

Creep test on marl in conventional three-axial devicewas carried out on 10 specimens with constant verticalload of 2.0 MPa over the period of 3 days and with4.0MPa load within the following 3 days. Three

specimens were tested without lateral pressure, thefourth one had lateral pressure of 1.0MPa, and threespecimens had a 2.0MPa lateral pressure. This shortcreep test offered some relative indicators of behaviourof the examined rock in time with influence of rotationsymmetrical lateral pressure.

Diagram in Figure 16. shows the creep test results ofthree-axially loaded cylindrical specimens with differentlateral pressure. Full lines indicate axial (vertical) strainand dash lines indicate tangential (horizontal) strain.Radial deformation with all the specimens after one dayof loading preserves the obtained level regardless theintensity of radial stress in the considered domain, whileaxial deformation indicates influence on lateral stress on

the flow of time deformations [16].






40

Značajno različito deformacijsko ponašanje laporca urazličitim pravcima ukazuje na to da se proračuninaponski zavisnih fenomena u laporcima i sličnimstijenama moraju sprovoditi na reološkim modelima kojiobuhvataju efekte transverzalne izotropije.

Testovi puzanja, sprovedeni na jednoaksijalnoopterećenim prizmatičnim uzorcima, ukazuju na to dalaporac, pri konstantnom jednoaksijalnom naponu pri-

tiska, pokazuje značajne vremenske deformacije -puzanje. U prvih dvadeset dana nakon opterećivanja,

priraštaj deformacija je nelinearan u odnosu na vrijeme.To je zona intenzivnog puzanja materijala u aksijalnompravcu - primarno puzanje. Nakon tog perioda, deforma-cije puzanja su manje i priraštaj deformacija je približnolinearan - sekundarno puzanje. Pritom, gradijent prirastadeformacije u toku vremena je veći kod uzorakaopterećenih većim naponom pritiska.

Bočni pritisak kod biaksijalno opterećenih pločastihuzoraka značajno mijenja deformacijsku sliku, kakopočetnih deformacija indukovanih naponskom promje-nom, tako i deformacija puzanja. Sa smanjenjem odnosa

vhk σ σ = od vrijednosti 1.0 prema (ka) jednoaksi-

jalnom stanju napona, prirast inicijalnih i vremenskizavisnih vertikalnih deformacija raste, što je posljedicaoslobađanja bočnog širenja uzorka. Uporedna analizavremenski zavisnih deformacija jednoaksijalno optere-ćenih prizmatičnih uzoraka i radijalnih deformacija okokružnog otvora (kod dvoosno opterećenih pločastihuzoraka - modela), ukazuje na to da nema bitne razlike uobliku i u ukupnoj veličini vremenskih deformacija. Ovoopet ukazuje na to da se rezultati puzanja dobijeni navrlo jednostavnim jednoaksijalnim testovima mogukoristiti u inženjerskoj praksi za ocjenu vremenskizavisnih deformacija meke stijene, pri znatno složenijemnaponskom stanju - kakvo je stanje napona okotunelskih otvora.

Efekat uticaja bočnog napona pritiska na aksijalnudeformaciju, pri konvencionalnom testu triaksijalne kom-presije, znatno je veći nego na uticaj bočnog napona kodbiaksijalnog opterećenja. Aksijalna deformacija, u uslovi-ma rotaciono simetričnog pritiska, može biti manja i do100% (pri hidrostatičkom stanju napona, k=1.0) odaksijalne deformacije, razvijene bez djelovanja bočnogpritiska.

transversal isotropic behaviour of material. Thesignificant differences in the behaviour of marl indifferent directions indicate that calculations of stress-dependent phenomena in marls and similar rocks shouldbe performed on rheological models which include theeffects of the transversal isotropy.

The creep tests of the uniaxially and biaxially loadedspecimens suggest that when marl is under constant

uniaxial stress state, significant time-dependentdeformations - creep are observed. In the first twentydays following the load application, the increment ofdeformations is non-linear in relation to time. This is azone of intensive creep of material in the axial direction -primary creep. After this period, the extent of creepdeformations is smaller and the increment ofdeformations is approximately linear - secondary creep.In addition, the gradient of deformation increment ishigher in the case of the specimens under highercompression pressure.

Lateral compression in the case of biaxiallycompressed plate specimens significantly changes thedeformation both the initial deformations induced by achange in stress and creep deformations. With the

decrease in value of the ratio vhk σ σ = from value of

1.0 towards uniaxial stress state, the increment of initialand time-dependent vertical deformations grows as aresult of the relief from lateral expansion of thespecimen. Comparative analysis of time-dependentdeformations in uniaxially compressed prismaticspecimens and radial deformations around the circularopening (in the case of biaxially compressed platespecimens-models) suggests that there is no relevantdifference in the shape and total extent of the time-dependent deformations. This indicates that the resultsobtained from the quite simple uniaxial compressiontests can be applied in the engineering practice for thepurposes of assessment of time-dependentdeformations in soft rock at far more complex stressstate, such as the stress state within the rocks aroundthe tunnel opening.

In the conventional three-axial compression test, theeffect of lateral strain on the axial deformation issignificantly larger compared to the effect of the lateralstrain in the case of the biaxial compression. Atrotationally symmetric pressure, the axial deformationcan be even up to 100% (at hydrostatic stress statek=1.0) smaller than the axial deformation that isgenerated without an impact of the lateral strain.

8 LITERATURA

REFERENCES

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[25] Bosman, J., Malan, D., and Drescher, K. 2000.„Time-dependent tunnel deformation atHartebeestfontein mine.” J. S. Afr. Inst. Min.Metall.,100(6), pp. 333–340.

[26] Cristescu, N.D. and Hunsche, U. 1998. Timeeffects in Rock Mechanics. John Willey & Sons.

[27] Cristescu, N.D. 2009. Time Effects in Rock Mecha-nics. Proceedings of the SEM Annual Conference,June 1–4, Albuquerque New Mexico USA ©2009Society for Experimental Mechanics Inc.




41

[28] Goodman, R. E. 1989. Introduction to rockmechanics,Wiley, New York. John Willey & Sons.

[29] Jaeger J.C. & Cook N.G.W., and Zimmerman R.W.2007. Fundamentals of Rock Mechanics, Chapmanand hall Ltd. And Science Paperbanks.

[30] Langer M., 1979, Rheologichal behaviour of rockmasses, Proc. 4th Int. Con. On Rock Mech.,Montreux, Vol.3., pp 29–62.

[31] Pan Y.W. and Dong J.J. 1991a. Time-DependentTunnel Convergence - I. Formulation of the Model,J.Rock Mech. and Mining Sci., pp. 469–475

[32] Pan Y.W. and Dong J.J, 1991b. Time-DependentTunnel Convergence - II. Advance Rate andTunnel-Support Interaction, J.Rock Mech. andMining Sci., pp. 475–488

[33] Pande G.N., Beer G. & Williams J.R. 1995.Numerical Methods in Rock Meshanics, John Wiley& Sons Ltd.

[34] Phienwej, N., Thakur, P. K., and Cording, E. J.2007. „Time-dependent response of tunnelsconsidering creep effect.” Int. J. Geomech., 7(4),p.p 296–306.

[35] Sakurai, S., 1978, „Approximate time-dependentanalysis of tunnel support structure consideringprogress of tunnel face.” Int. J. Numer.Analyt.Meth. Geomech., 2, 59–175.

[36] Tomanovic, Z. 2006. Rheological model of soft rockbased on test on marl. Int. J. Mechanics of Time-Dependent Materials, Springer, pp. 135–154.

[37] Tomanovic, Z. 2007. Reološki model puzanja

matriksa meke stijene / Rheological model ofmatrix of soft rock creep, Građevinski materijali ikonstrukcije / Building matrilas end structure, vol.50, br. 1-2, str. 3-19

[38] Tomanovic, Z. 2009. Influence of Ko on creepproperties of marl. Int. J. Acta GeotechnicaSlovenica, pp. 14-29.

REZIME

TESTIRANJE FENOMENA PUZANJA MEKE STIJENE

Zvonko TOMANOVI Ć

Test puzanja karakteriše konstantno opterećenje podkojim se materijal s vremenom deformiše. Zbog toga jeza korektno izvođenje testa puzanja veoma važnoodržavanje - kontrolisanje konstantnog naponskogstanja koje je aplicirano na uzorak. Naime, uticaj naponapritiska na veličinu deformacije puzanja izrazito jenelinearan, pa zbog toga nivo napona, odnosnoeventualne varijacije napona u toku vremena imaju velikiuticaj na prirast vremenski zavisnih komponentideformacija. Radi toga, u toku testa puzanja – kojinekada traje i godinama – treba obezbijediti da varijacijenapona budu u granici od ±1%. U ovom radu predstavljase konstrukcija aparature s „mrtvim” opterećenjem zaodržavanje konstantnog naponskog stanja u toku dugogperioda (dana, nedjelja ili godina), kapaciteta od750 kN i1500 kN, kao i mjerenje sile i deformacija u toku testapuzanja. Testovi puzanja sprovedeni su na uzorcimalaporca, u trajanju do jedne godine. Predstavljeni surezultati jednoaksijalno opterećenih prizmatičnih uzoraka

dimenzija 15x15x40 cm i rezultati testa puzanjabiaksijalno opterećenih pločastih modela dimezija60x60x10 cm, s kružnim otvorom u sredini. Takođe,prikazani su i rezultati testa puzanja u triaksijalnomaparatu na uzorcima dimenzija d/h = 5.4/108 cm.

ključne riječi: puzanje, aparatura za opterećivanje,laporac, mjerenje sile i deformacija, meka stijena

SUMM АRY

TESTING OF CREEP PHENOMENA ON SOFT ROCK

Zvonko TOMANOVIC

Creep test involves deformation of material with timeunder a constant stress. Hence, for proper conducting ofthe creep test it is highly important to maintain, i.e.constant control the stress applied to a specimen.

Namely, impact of the compression on the extent ofdeformation is distinctively non-linear; therefore thestress level or eventual variations of stress over timelargely affect the increment of time-dependentcomponents of deformation. Therefore, it is necessary toensure that variations of stress are kept within limits of±1% during the creep test the duration of whichsometimes may last for several years. This paperpresents structure of the dead-weight creep apparatusfor maintaining constant stress over long time interval(day, week or year) with a capacity of 750kN and1500kN, as well as for measuring of force anddeformations at creep test. The creep tests have beenconducted on marl specimens in duration of up to oneyear. The creep test results are presented for the

uniaxially compressed prismatic specimens ofdimensions 15x15x40cm and biaxially compressed platemodels of dimensions 60x60x10cm, with circularopening at centre.

Key words: creep, load-application apparatus, marl,force and deformation measurements, soft rock




42






44

je uveo indeks višekriterijumskog rangiranja, koji seodređuje na osnovu blisjosti idealnom rešenju.

Opricović i Tzeng [13] upoređivali su osnovnekarakteristike VIKOR i TOPSIS metoda u svim koracimarešavanja problema: proceduralna baza, normalizacija,agregacija i konačno rešenje. Kasnije je Opricovićproširio VIKOR metod za rešavanje rasplinutihvišekriterijumskih problema s konfliktnim i nekonfliktnim

kriterijumima i razvio metod VIKOR-F [14]. Metod VIKORviše puta je korišćen za višekriterijumsko rangiranjealternativa prilikom rešavanja mnogih problema ugrađevinarstvu, hidrotehnici i saobraćaju, kao i u drugiminženjerskim oblastima. Pored ove dve metode, uliteraturi postoji još metoda za višekriterijumskodonošenje odluka (AHP, PROMETHEE, ELECTRE i dr.).

Wang i Elhang [17] predložili su fuzzy TOPSISmetod, zasnovan na alfa presecima rasplinutih skupovaza rešavanje problema upravljanja rizikom kod mostova.Za svaku alternativu i izabrani alfa nivo, definisali sunelinearni program (NLP) s gornjom i donjom vrednošćurelativne bliskosti od NIS kao funkcijama cilja i u većdefinisanim gornjim i donjim vrednostima kaoograničenjima. Na taj način, relativne bliskosti

posmatrane su kao rasplinuti brojevi, a kasnije, posledefazifikacije, rangirane su alternative. U stranoj Idomaćoj literaturi postoje brojni primeri primene TOPSISmetode s fiksnim i rasplinutim brojevima u svimpodruč jima građevinarstva i realizacije investicionihprojekata za rangiranje alternativa ili subjekata imajući uvidu propisane kriterijume. Kraći prikaz jednog broja tihradova dali su autori u njihovom ranijem radu [16].

Procena rizika objekata (mostova, zgrada i ostalihobjekata) najčešće se koristi za određivanje optimalnogplana ili rangiranja održavanja objekata u odnosu narizik. Ovaj problem proučavali su mnogi autori, a uliteraturi postoje različiti metodi procene rizika. Naprimer, Adey, Hajdin i Brühwiler [1] predstavili su pristup

određivanju optimalnog plana održavanja mostova,baziranog na rizicima koje je izazvalo više hazarda.Wang i Ehlang [18] predložili su pristup grupnomdonošenju odluka za procenu rizika, koristeći rasplinutiTOPSIS metod.

Mnogi radovi u kojima je reč o oceni stanja,održavanju i sanaciji građevinskih objekata i naseljapublikovani su u zbornicima radova s međunarodnihkonferencija, čiji je editor Folić [9], [10].

U ovom radu razmatra se problem višekriterijumskograngiranja objekata za rekonstrukciju na osnovudefinisanih kriterijuma, korišćenjem modifikovanerasplinute TOPSIS procedure koju su predložili autori, akoja je detaljno prikazana u radu [16]. U ovom metodu,

svi ulazni podaci predstavljeni su kao trougaoni rasplinutibrojevi. Za ove brojeve i njihove proizvode određene sugeneralisane očekivane vrednosti, varijanse, standаrdnedevijacije i koeficijenti varijacija. Ove vrednosti, dalje,korišćene su u matematičkim formulama za određivanjerelativnih rastojanja svake alternative do PIS i NIS zanjihovo rangiranje. Predložena procedura je opštija odprocedure u kojoj se koriste fiksni brojevi i donosiocuodluka pruža realnije podatke za donošenjenajprihvatljivije odluke.

criteria. He introduced the multiple criteria ranking indexbased on the particular measure of closeness to theideal solution.

Opricović and Tzeng [13] compared main features ofVIKOR and TOPSIS metods in all steps of problemsolution: procedural basis, normalization, aggregationand final solution. Opricović later extended VIKORmethod for solving fuzzy multiple criteria problems with

conflicting and non conflicting criteria and developedVIKOR-F [14] . VIKOR method has been used manytimes for multiple criteria ranking of alternatives forsolving many problems in civil, hydro technical andtransportation engineering and other branches of practice as well. Besides these two methods, there aremore methods for multiple criteria decision making(AHP, PROMETHEE, ELECTRE, etc.) in the literatureconsidering this field of research.

Wang and Elhang [17] proposed fuzzy TOPSISmethod based on alfa level sets with application to thebridge risk management. For every alternative andchosen alfa level, they formulated nonlinear programs(NLP) with lower and upper value of relative closenessto NIS as the objective functions and with prescribed

lower and upper values as the constrains. In such a waythese relative closeness are considered as fuzzynumbers, and then after defuzzification, the alternativesare ranked according to these closeness. In the foreignand domestic literature there is large number ofexamples of application of the TOPSIS method in allarea of civil engineering and construction projectrealization for ranking alternatives or subjects related tothe prescribed criteria. Short review of these works ispresented in the author's work [16].

The risk assessment of an object (bridge, building,etc) is usually performed to determine the optimalscheme or rank order of the object maintenance. Thisproblem has been investigated by numerous authors

and there are different methods for the risk assessment.For instance, Adey, Hajdin and Brühwiler [1] presentedrisk-based approach to the determination of optimalinterventions for bridges affected by multiple hazards.Wang and Ehlang [18] proposed a fuzzy group decisionmaking approach for the risk assessment using fuzzyTOPSIS method.

Numerous papers related to the assessment,maintenance and rebuilding of structures andsettlements are given in the proceedings of internationalconferences, edited by Folić [9],[10].

This paper deals with a problem of multiple criteriaranking of objects for reconstruction against prescribedcriteria using modified fuzzy TOPSIS procedure

proposed by authors in the paper [16]. In this method allinput data are presented as probabilistic triangular fuzzynumbers. Generalized expected values, variances,standard deviations and coefficients of variations arefound for these fuzzy numbers and their products. Thesevalues are, further, used in mathematical formulas todetermine relative closeness of every alternative to thePIS and NIS for their ranking. This proposed procedureis more general than the procedure based on crisp dataand gives to the decision maker more realistic data tomake the most acceptable decision.




45

2 DEFINICIJA PROBLEMA

U ovom radu razmatra se neka firma ili institucija(vlasnik) koja je odgovorna za održavanje n objekata(zgrada, mostova i drugih) koji su označeni sa A1, A2,…, Am. Da bi se smanjile posledice rizika koje utiču nasigurnost, funkcionalnost, održivost, raspoloživost,uticaje okoline i druge bitne faktore, neophodno je uložitiodređenu količinu novca za održavanje tih objekata.Raspoloživa količina novca obično nije dovoljna zaodržavanje svih objekata, pa je zbog toga neophodnoobjekte rangirati prema nivou rizika, te novac uložiti uobjekte shodno listi rangiranja. Pomenuti faktori, koji sezovu kriterijumi , označeni su sa C 1, C 2,…,C n, dok objektipredstavljaju alternative višekriterijumskog odlučivanja(MCDM). Svaku alternativu Ai u odnosu na kriterijum C i numerički su ocenili eksperti vrednošću f ij (i = 1,2,...,m; j

=1,2,...,n). Ove vrednosti jesu elementi matriceodluč ivanja, koja je označena sa F= [ f ij]m×n.

Skup kriterijuma Ω sadrži dva disjunktna skupa Ωb iΩc , tj.

2 DEFINITION OF THE PROBLEM

A firm or institution (owner) which is responsible forthe maintenance of n objects (buildings, bridges or otherobjects) A1, A2, Am is considered in this paper. To reduceconsequences of a risk that influence safety,functionality, sustainability, availability, environmentaland other important factors, a corresponding amount ofmoney should be invested in the maintenance of theseobjects. The available amount of money usually isinsufficient for all objects or projects, so that they shouldbe ranked according to the risk rating, and the moneyshould be invested in the objects according to this ranklist. The mentioned factors are named as criteriadenoted by C 1, C 2,..., C n, while the objects representalternatives for multi-criteria decision making (MCDM).Each alternative Ai is numerically evaluated by expertswith respect to the criterion C j by values f ij (i = 1,2,...,m; j

= 1,2,...,n). These values are elements of the decisionmatrix denoted by F= [ f ij]m×n

The set of criteria Ω contains two disjunctive subsetsΩb and Ωc , i.e

Ω = (C1,C2,...,Cn) = (Ω b U Ωc), (Ωc ∩ Ω b) = Ø. (1)Podskup Ωb predstavlja dobiti ili kriterijume s

povoljnim efektima koje treba maksimizovati, dokpodskup kriterijuma Ωc predstavlja troškove ili kriterijumes nepovoljnim efektima koje treba minimizovati.

Svaki kriterijum C j eksperti ocenjuju relativnomtežinom ili faktorom znač ajnosti w j ( j = 1,2,…,n). Ovevrednosti formiraju vektor težina w = [w j]1×n. Ciljrešavanja problema jeste da se odredi najprihvatljivija ilinajbolja alternativa – Ac koja zadovoljava sve kriterijumei koja je najbliža pozitivnom idelanom rešenju, anajudaljenija od negativnog idealnog rešenja, kao i da sealternative rangiraju prema navedenom pravilu.

Idealno pozitivno rešenje F * sadrži vrednosti f ij koje

predstavljaju maksimume kriterijuma dobiti i minimumekriterijuma troškova to jest

The subset of criteria Ωb represents benefits orcriteria with favourable effects that should be maximised,while subset of criteria Ωc represents costs or criteriawith unfavourable effects that should be minimized in theprocedure.

Every criterion C j is assessed by experts with relativeweight values or factors of importance w j ( j = 1,2,…,n).These values form the vector of weights w = [w j]1×n. Thegoal of the problem solution is to find the most preferableor the best (compromise) alternative Ac that satisfies allcriteria together and which is closest to the positive idealsolution and farthest to the negative ideal solution, andrank alternatives according to this rule as well.

The positive ideal solution F * contains the values f ithat are maximal for the benefit criteria and minimal forthe cost criteria, i.e.

),(),,(...,.. minmax***

1

*

cij jbij jni i f i f f f f F Ω∈Ω∈== . (2)

Idealno negativno rešenje F - sadrži vrednosti f ij koje

odgovaraju minimumima kriterijuma dobiti imaksimumima kriterijuma troškova to jest

The ideal negative solution F - contains values f

i

that are minimal for the benefit criteria and maximal forthe cost criteria, i.e.

),(),,(...,..maxmin

1 cij jbij jni i f i f f f f F Ω∈Ω∈== −−−−. (3)

3 TOPSIS PROCEDURA SA FIKSNIM BROJEVIMA

Ako su elementi matrice odlučivanja f ij (i = 1,2,...,m; j= 1,2,...,n) i koeficijenti značajnosti ili težine kriterijuma w j

( j = 1,2,…,n), fiksni ili nerasplinuti brojevi, onda seprimenjuje TOPSIS procedura s fiksnim brojevima, kojase izvršava u sledećim koracima.

1. Normalizacija. Pošto kriterijumi mogu imati različitaznačenja i različitu prirodu, elementi matrice odlučivanjaizražavaju se različitim dimenzionalnim merama i skalama

3 TOPSIS PROCEDURE WITH CRISP NUMBERS

If elements of the decision matrix f ij (i = 1,2,...,m; j =1,2,...,n) and coefficients of importance or weights ofcriteria w j ( j = 1,2,…,n), are crisp or non fuzzy numbers,then the TOPSIS procedure with crisp numbers isapplied, which performs in the next steps.

1. Normalization. Since the criteria may havedifferent meanings and nature, then elements of thedecision matrix are expressed by different dimensional




46

vrednosti. Stoga, treba izvršiti normalizaciju elemenatamatrice odlučivanja F. U literaturi postoji više predlogaza normalizaciju, a ovde će biti prikazana dva koja senajčešće primenjuju.

Prema prvom postupku, za svaki kriterijum C j ( j =

1,2,…,m) odredi se maksimalna vrednost

measures and scales of values. Because of that, thenormalization of elements of the decision matrix F

should be performed to obtain dimensionless values.There are several proposals for this normalization inliterature and here will be presented two of them that aremost frequently applied.

According to the first proposal for every criterionC j ( j = 1,2,…,m) maximal value is determined

),...,2,1;,...,2,1(max* n jmi f f iji

j === (4)

i s tom vrednošću podele sve vrednosti f ij u koloni C jmatrice F. Na taj način, dobijaju se normalizovane ibezdimenzionalne vrednosti aij koje sačinjavajunormalizovanu matricu odlučivanja A = [aij]

and with this value all values f ij in the column C j of thematrix F are divided. Thus, normalized and nondimensional values aij that compose normalized decisionmatrix A = [aij] are obtained

).,...,2,1;,...,2,1(,/ * n jmi f f a jijij === (5)

Drugi postupak naziva se vektorski postupak i unjemu se za svaki kriterijum nađu dužine odgovarajućihvektora po formuli

The second proposal is named a vector procedure inwhich lengths of corresponding vectors are determinedfor every criterion

),...,2,1(,... 22221* n j f f f f mj j j j =+++= . (6)

Kao i u prethodnom slučaju, sa ovim vrednostimadele se elementi matrice odlučivanja F i tako dobijajuelementi normalizovane matrice A.

2. Određ ivanje težinske matrice C. Svaki elementnormalizovane matrice A množi se sa odgovarajućimtežinskim koeficijentom ili koeficijentom značajnostikriterijuma w j i tako se dobiju elementi c ij težinskematrice C = [cij]

As in the previous case, elements of the decisionmatrix F are divided by these values and obtainedelements aij of the normalized decision matrix A.

2. Determination of the weighted matrix C. Everyelement of the normalized matrix A is multiplied by thecorresponding weighted coefficient or coefficient osignificance w j to obtain elements c ij of the weightedmatrix C = [cij]

).,...,2,1:,...,2,1(, n jmiwac jijij === (7)

3. Određ ivanje pozitivnog idealnog rešenja (PIS) i

negativnog idealnog rešenja (NIS)Za svaku alternativu Ai određuju se komponente

*

ic

pozitivnog idealnog rešenja i _

jc negativnog idealnog

rešenja prema sledećim formulama

3. Determination of the positive ideal solution (PIS)

and negative ideal solution (NIS)For every alternative Ai are determined components

*

ic

of positive ideal solution and components _

jc of

negative ideal solution according to the next formulas

.,...,2,1;,...,2,1;∈:minor ∈:max _ * n jmiC ccC cc c jij

j jb jij

ji ==== Ω Ω (8)

.,...,2,1;,...,2,1;∈:maxor ∈:min _ * n jmiC ccC cc c jij

j jb jij

ji ==== Ω Ω (9)

4. Određ ivanje udaljenosti i relativnih bliskostii

alternativa Ai pozitivnom idealnom (PIS) i negativnomidealnom rešenua (NIS)Za svaku alternativu Ai određuje se distanca od

pozitivnog idealnog rešenja*

i D i negativnog idealnog

rešenja−*

i D prema sledećim formulama

4. Determination of distances and relative closeness

of the alternatives Ai to positive ideal solution (PIS) annegative ideal solution. (NIS)

For every alternative Ai the distances*

i D from the

positive ideal solution and−*

i D from the negative ideal

solution are determined by the following formulas

;)(

2/1

1

2**⎥⎦

⎤⎢⎣

⎡−= ∑

=

m

j

iiji cc D ;,...,2,1;,...,2,1;)(

2/1

1

2 n jmicc Dm

j

iiji ==⎥⎦

⎤⎢⎣

⎡−= ∑

=

−− (10)




47

i relativne bliskosti *

i RC pozitivnom idealnom rešenju i−i RD i negativnom idealnom rešenju

and relative closeness *

i RC to positive ideal solution

andi RC and to negative ideal solution

),/( ***

iiii D D D RC += .,...,,;,...,,);/( * n jmi D D D RC iiii 21=21=+= (11)

.* 1=+ ii RC RC (12)

Ove relativne bliskosti nazivaju se još i koeficijenti bliskosti alterantive Ai pozitivnom idealnom rešenju inegativnom idealnom rešenju.

Alternative se rangiraju prema ovim koeficijentima. Alternative s manjom relativnom bliskosšću

*

i RC pozitivnom idealnom rešenju, i većim relativnom

bliskosšću _

i RC negativnom idealnom rešenju, bolje su

rangirane. Najbolje rangirana alternativa jeste ona kojaima najmanji koeficijent bliskosti idealnom pozitivnom

rešenju*

i RC .

3.1 Primer

Radi lakšeg razumevanja ove procedure, razmotrićese jedan jednostavan primer. Neka postoje tri alternative A1, A2 i A3 i dva kriterijuma C 1 i C 2, i neka se kriterijum C 1odnosi na trošak, a kriterijum C 2 na korisnost (dobit),tako da su skupovi kriterijuma

The relative closeness are named coefficients ocloseness of the alternative Ai to the positive idealsolution and negative ideal solution respectively. Thealternatives are ranked according to these coefficients.

Alternatives with the smaller relative closeness*

i RC to

the positive ideal solution and greater relative closeness _

i RC to the negative ideal solution are better ranked.

The best ranked alternative has the smallest coefficient*

i RC .

3.1 Example

For the easiest understanding of this procedure, onesimple example will be considered. Let exists threealternatives A1, A2, A3 and two criteria C 1, C 2, and letthe criterion C 1 is related to the cost, and criterion C 2 tothe benefit (profit), so that the sets of criteria are

Ωb = C 2 , Ωc = C 1, (Ωc∩ Ωb) = Ø.

Neka su procenjene vrednosti matrice odlučivanja Let assess values of the decision matrix

C 1 C 2

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡

=00.300.400.100.3

00.200.1

F

3

2

1

A A

A

.

Kriterijum C 1 se minimizuje, a kriterijum C2 semaksimizuje, tako da su prema (3) elementi pozitivnogidealnog rešenja (PIS)

The criterion C 1 is minimised, while criterion C 2 ismaximised, so the elements of positive ideal solution(PIS), according to (3), are

00.1)00.4,00.3,00.1(min#

1 == f , 00.3)00.3,00.1,00.2(max#

2 == f ,

i negativnog idealnog rešenja (NIS) and negative ideal solution (NIS)

00.4)00.4,00.3,00.1(max1 ==− f , 00.1)00.3,00.1,00.2(min2 ==− f .

Ova rešenja predstavljaju dve zamišljene idealne

alternative, PIS - [ ]003001= ..* A i NIS - [ ].00.100.4=− A

U koordinatnom sistemu kriterijuma C 1 i C 2, na slici1, sve ove alternative prikazane su kao tačke.

Pod uslovom da su koeficijenti težina (značajnosti)oba kriterijuma isti i da su vrednosti elementata matriceF za oba kriterijuma izraženi u istim jedinicama merenije neophodno vršiti normalizaciju ovih vrednosti.Određuju se udaljenosti od tačaka A1, A2 i A3, koje

predstavljaju alternative od tačaka-*

i A A , koje

predstavljaju pozitivno (PIS) i negativno (NIS) idealnorešenje respektivno se određuju u sledećem koraku..

These solutions represent imaginary ideal alterna-

tives, PIS - 003001= ..* A and NIS [ ].00.100.4=− A In the coordinate system of criteria C 1 and C 2, shown

in Fig. 1, all the alternatives are presented as points.Provided that the coefficients of weights (coefficients

of importance) for the both of criteria are the same andthat the values of elements of the matrix F for both ofcriteria are expressed in the same units of measure, it isunnecessary to perform normalisation of these values.Distances of the points A1, A2 and A3, that represent

alternatives, from the points *i

− A A , that represent

positive ideal solution (PIS) and negative ideal solution(NIS) respectively are determined in the next step.




48

C 2

A* D3* A3

3.00 D1

* D2*

2.00 A1 D3

-

D1-

1.00 A2 - A--

D2-

0.00 C 1

0.00 1.00 2.00 3.00 4.00 5.00

Slika 1. Grafi č ki prikaz alternativa i kriterijumaFigure 1. Graphical presentation of the alternatives and criteria

Udaljenosti*

i D (i = 1,2,3) alternativa od pozitivnogidealnog rešenja (PIS) jesu:

The distances*

i D (i = 1,2,3) of the alternatives fromthe positive ideal solution (PIS) are according to (10):

00.1])00.300.2()00.100.1[( 2/122*

1 =−+−= D ,

82.2])00.300.1()00.100.3[( 2/122*

2 =−+−= D ,

00.3])00.300.3()00.100.4[( 2/122*

3 =−+−= D .

Udaljenosti−*

id (i = 1,2,3) alternativa od negativnog

idealnog rešenja (NIS) jesu:

The distances _

i D (i = 1,2,3) of the alternatives from

the negative ideal solution (NIS) are according to (10):

16.3])00.100.2()40000.1[( 2/122

1 =−+−=− D ,

00.1])00.100.1()00.400.3[( 2/122

2

=−+−=− D ,

00.2])00.100.3()00.400.4[( 2/122

3 =−+−=− D .

Na kraju, sračunavaju se relativne bliskostialternativa od PIS *

i RC (i = 1,2,3) i relativne udaljenosti

alternativa od NIS-

i RC (i = 1,2,3):

At the end, are calculated relative closeness to the

alternatives*

i RC (i =1,2,3) from PIS and-

i RC (i =1,2,3)

from NIS, according to (11), :

,.)../(.* 240=163+001001=1 RC ,.)../(. 760=163+001163=1 RC

,.)../(.* 740=001+822822=2 RC ,.)../(. 260=001+822001=2 RC

,.)../(.* 600=002+003003=3 RC ..)../(. 400=002+003002=3 RD

Na osnovu ovih rezultata, može se zaključiti da

alternativa A1 ima najmanju udaljenost 00.1*

1 = D inajmanju relativnu bliskost PIS pozitivnom idealnom

rešenju 240=1 .* RC koje je predstavljeno tačkom A*.

Ova alternativa ima najveću udaljenost 16.31 =− D i

najmanju relativnu bliskost NIS 76.0-1 = RC od NIS, koje

je predstavljeno tačkom A- na slici 1. Prema tome,alternativa A1 je najbliža ili „najsličnija” pozitivnomidealnom rešenju A*, pa je zbog toga – najprihvatljivija. Ako se izvrši rangiranje prema relativnoj udaljenosti odpozitivnog idealnog rešenja, onda je redosled alternativa A1, A3, A2.

From these results may be concluded that alternative

A1 has the smallest distance 00.1*

1 = D and smallestrelative closeness from PIS

240=1 .* RC , which is

represented by the point A*. This alternative has thelargest PIS distance 163=1 .* RC

and the largest relative

closeness to NIS 76.01 =− RC , that is represented by the

point A- in Fig. 1. Therefore, the alternative A1 is thenearest or "most similar" to the positive ideal solution A*,and because of that it is most acceptable. If alternativesare ranked according to the relative distance ofalternatives from the positive ideal solution, then order ofalternatives is A1, A3, A2.






50

Rasplinuti skup je konveksan ako su mu svi α -

preseci α A konveksni skupovi.

Ako univerzalni skup predstavlja skup realnih brojeva R, čiji su elementi x predstavljeni brojnom pravom, onda

se rasplinuti skup R A ∈~

naziva rasplinuti (fuzzy ) broj ,

s funkcijom pripadnosti ],1,0[:)(~ → R x A

ako

ispunjava sledeće uslove (Dubois and Prade, [5]):• A

~ je normalan broj, što znači da postoji broj

R x ∈0 za koji je ;1)( 0~ = x A

• A~

je rasplinuto konveksan broj, tj.

),(),(min))1(( ~~~ y x y x A A A

λ λ ≥−+ za sve

];1,0[,, ∈∈ λ R y x

• A~

je odozgo semikontinualan broj (tj.

])1,([1~ α µ −

A zatvoren je za sve ]),1,0[∈α

• Osnova rasplinutog broja A~

))~

( ASupp

zatvoren je skup za koji je .,0)(~ R x x A ∈>µ

• Aα – je α presek rasplinutog (fuzzy) broja jenerasplinuti broj

• 10,)(| ~ ≤<≥∈= α α α x R x A A

, koji

predstavlja zatvoreni interval poverenja

The fuzzy set is convex one if all its α - cuts α A are

convex sets.If universe of discourse is the set of real numbers R,

whose elements x are represented by the numeric

straight line, then the fuzzy set R A ∈~

is named a fuzzynumber , with the membership function

],1,0[:)(~ → R x A

µ if satisfies the following conditions

(Dubois and Prade, [5]):

• A~

is a normal number, which means that exists a

number R x ∈0 for which is ;1)( 0~ = x A

• A~

is fuzzy convex number, i.e.

),(),(min))1(( ~~~ y x y x A A A

µ µ λ λ µ ≥−+ for all

];1,0[,, ∈∈ λ R y x

• A~

is upper semi continuous (i.e. ])1,([1~ α µ −

Ais

closed for all ]),1,0[∈α

• The support of A~

( ))~

( ASupp is bounded for

which is .,0)(~ R x x A

∈>

• Aα is α cut of the fuzzy number A~

, that is a crispnumber

• 10,)(| ~ ≤<≥∈= α α α x R x A A

, which

represents a closed interval of confidence

,10)],(),([ ≤<= α α α α ul A A A (15)

)(i)( α α ul A A jesu donja i gornja granica ovog

intervala. Par funkcija )(i)( α α ul A A predstavlja

parametarsku prezentaciju rasplinutog broja .~ A

)(i)( α α ul A A are lower and upper limits of this

interval. The pair of functions of this interval )(α l A and

)(α u

A expresses a parametric presentation of the

fuzzy number .~

A

.)(:sup)(,)(:inf)( ~ α α α α ≥∈=≥∈= x R x A x R x A Au Al

(16)

Zavisno od funkcije pripadnosti, postoji više tipovarasplinutih brojeva: trougaoni, trapezoidni, broj oblika S iπ i drugi. U teoriji i praksi, najčešće se zbog linearnostifunkcija pripadnosti koriste trougaoni i trapezoidnirasplinuti brojevi. U ovom radu se koristi trougaonirasplinuti broj, prikazan na slici 1. Ovaj broj se običnoprikazuje s tri karakteristične vrednosti: donjom al ,

modalnom am (za koju je µ(am)=1) i gornjom au , tj.

There are several types of fuzzy numbers dependingof membership function: triangular, trapezoidal, S andπ shape and others. In the theory and practicalapplications triangular and trapezoidal fuzzy numbersare used most frequently. Triangular fuzzy number,shown in Fig. 1, is used in this paper. This number isusually represented by three characteristic values: lower

al , modal am (for which is µ(am) =1) and upper au, i. e.

),,(~uml aaa A = . (17)

Funkcije pripadnosti ovog broja jesu: The membership functions of this number are:

.za0)(

,za)/()()(

,za)/()()(

,za0)(

~

~

~

~

u A

ummuu A

ml l ml A

l A

a x x

a xaaa xa x

a xaaaa x x

a x x

≥=

≤≤−−=

≤≤−−=

≤=

µ

µ

µ (18)








53

`2

~

~2

)()(

)(

)~

( U

e

R

A

R

AU

e xdx x

dx x x

AV −=∫

∫µ

µ

. (30)

Za trouglasti rasplinuti broj ),,(~

uml aaa A = ove

vrednosti su

For the triangular fuzzy number ),,(~

uml aaa A =

these values are

,3/)()~

( uml

U

e aaa A x ++= (31)

.18/)()~

( 222

umul ml uml

U

e aaaaaaaaa AV −−−++= (32)

Ako je raspodela verovatnoće P ( x) trougaona, tako

da je proporcionalna funkciji pripadnosti )(~ x A

If the probability distributions function P ( x) istriangular, so that it is proportional to the membership

function )(~ x A

),()( ~ xk x P A

= (33)

gde je k factor proporcionalnosti, onda su:

− generalisana očekivana vrednost rasplinutogbroja

where k is factor of proportionality then is:

− generalized expected value

∫

∫=

R

A

R

AT

edx x

dx x x

A x2

~

2~

))((

))((

)~

(µ

µ

, (34)

− generalisana varijansa − generalized variance

2

2~

2~

2

))~

((

))((

))((

)~

( A x

dx x

dx x x

AV T

e

R A

R

AT

e −=

∫

∫

µ

µ

. (35)

Za trouglasti rasplinuti broj ),,(~

uml aaa A = ove

vrednosti su

For the triangular fuzzy number ),,(~

uml aaa A =these values are

,4/)2()~

( uml

T

e aaa A x ++= (36)

.80/)424343()~

( 222

umul ml uml

T

e aaaaaaaaa AV −−−++= (37)

5 MODIFIKOVANA RASPLINUTA (FUZZY) TOPSISPROCEDURA

Elementi rasplinute matrice odlučivanja F~

su tro-

ugaoni rasplinuti brojevi ),,(~ )(

.

)()(

,

u

ij

m

ij

l

ijij f f f f = , tako

da se ova matrica može prikazati pomoću tri matrice s

fiksnim (nerasplinutim) elementima ).,,(~

uml FFFF=Rasplinuta TOPSIS procedura izvršava se u nekolikokoraka koji su objašnjeni u ovom radu, uz predloženumodifikaciju. Ovi koraci su: normalizacija, računanjegeneralisane očekivane vrednosti i standardne devija-cije, rangiranje alternativa i izbor najbolje alternative.

MODIFIED FUZZY TOPSIS PROCEDURE

Elements of the fuzzy decision matrix F~ are

triangular fuzzy numbers ),,(~ )(

.

)()(

,

u

ij

m

ij

l

ijij f f f f = , so

that this matrix can be expressed by three crisp matrices

).,,(~

uml FFFF= Fuzzy TOPSIS procedure performs in

several steps that will be explained in this paper withsome proposed modification. These steps are:normalization, calculation of generalized expectedvalues and standard deviations, ranking alternatives andchoice of the best alternative.




54

5.1 Normalizacija

Normalizacija se i ovde vrši iz istih razloga iz kojih seto čini i u TOPSIS metodi s fiksnim (nerasplinutim)brojevima – da bi se dobile bezdimenzionalne vrednosti

u matrici odlučivanja F~

. Međutim, zbog rasplinutostinjezinih elemenata, u literaturi postoji nekoliko predlogaza normalizaciju (Wang i Elhang, [17]). Ovde će se

koristiti metod koji su predložili Ertugrud i Karakasagly[6]. Normalizovane vrednosti elementa ij f

~ rasplinute

(fuzzy) matrice odlučivanja F~

označeni su saija~ , i one

sačinjavaju normalizovanu rasplinutu matricu

odlučivanjaA~

i sračunavaju se po sledećoj formuli

5.1 Normalization

Normalization is performed here due to the samereasons as in TOPSIS method with the crisp numbers to

obtain dimensionless values in the decision matrix F~

.However, due to fuzziness of its elements, there areseveral proposals for the normalization (Wang andEhlang, [17]). Here, a method which is proposed by

Ertugrud and Karakasagly [6] is used. Normalized values

of elements ij f ~

are denoted as ija~ , and they

constitute the normalized fuzzy matrix A~

and they arecalculated by the following formula

;,...2,1;,...,2,1;)/,/,/(~ )*()*()*()(n jmi f f f f f f a

u

i

u

ij

u

i

m

ij

u

i

l

ijij === (38)

gde za svaki kriterijum C i where for every criterion C i

.,...,2,1,)(max)*(mi f f

u

ij j

u

i == (39)

5.2 Određivanje karakterističnih vrednostielemenata težinski normalizovane matrice

odlučivanja C~

Elementi ijc~ težinske normalizovane matrice

odlučivanja C~

računaju se kao proizvodi dva rasplinuta

trouglasta brojaija~ i težine jw koja u većini slučajeva

predstavlja koeficijent znač ajnosti kriterijuma C j

5.2 Determination of characteristic values of the

weighted normalized decision matrix C~

Elements ijc~ of the weighted decision matrix C~

are

calculated as a product of two fuzzy numbersija~ and

the weight jw~ , which in many cases represents

coefficient of significance of the alternative C j

;,...,2.1;,...,2,1,~~

n jmivac iijij === (40)

Rasplinuti brojevi ija~ i jw~ se mogu prikazati prema (17) Fuzzy numbers ija~ i jw~ may be shown according to

(17)

),,,(~ )()()( u

ij

m

ij

l

ijij aaaa = ),,,(~ )()()( u

i

m

i

l

ii wwww = i =1,2,..,m; j =1,2,...n.

Lako se može zaključiti, iz pravila o množenjurasplinutih brojeva (23), da proizvod dva trougaonarasplinuta broja nije trougaoni rasplinuti broj, tako da

elementi ijc~ matrice C~

nisu trouglasti rasplinuti brojevi.

Međutim, mnogi autori pretpostavljaju, radi uprošćenjaprocedure, da ovi elementi jesu trouglasti brojevi i

sračunavaju elemente težinske fuzzy matrice C~ prema

sledećoj formuli

It is easy to conclude from the rule of multiplication offuzzy numbers (23), that product of two fuzzy triangularnumbers is not a triangular fuzzy number, hence

elements ijc~ of the fuzzy matrix C~

are not triangular

fuzzy numbers. However many authors suppose, due tosimplicity of the procedure, that these numbers aretriangular ones and calculate elements of this matrix bythe simplified formula

.,...,2,1;,...,2,1),~~,~~,~~(~ )()()()()()( n jmiwawawac u

i

u

ij

m

i

m

ij

l

i

l

ijij === (41)

U svom prethodnom radu [16], autori su izveliobrasce za tačno određivanje ovih rasplinutih brojevakoji nisu trougaoni, i predložili proceduru s generalisanim

očekivanim vrednostima ije i varijansama vij proizvoda

rasplinutih brojeva jijij wac ~~~ =

In the earlier paper written by these authors [16], a

procedure with the generalized expected values ije and

variances vij of the fuzzy numbers products

jijij wac ~~~ = has been proposed.




55

.,...,2,1;,...,2,1);~(),~( n jmicV vc xe ijijijeij ==== (42)

U toj proceduri, brojevi ijc~ tretiraju se kao slučajni

fuzzy događaji koji s jedne strane imaju odgovarajućuraspodelu verovatnoće događanja, a s druge funkciju

pripadnosti rasplinutom (fuzzy) skupu A~

.

Ove vrednosti su elementi matrica E i V respektivno,a računaju se prema formulama izvedenim upomenutom radu [16], zavisno od izabrane raspodeleverovatnoća fuzzy događaja, koje mogu biti uniformne ilitrougaone (proporcionalne).

In this procedure, the fuzzy numbers ijc~ are assumed

as fuzzy events that have a corresponding probabilitydistribution as well as membership function to the fuzzy

set A~

.

These values are elements of matrices E and Vrespectively and they are calculated by the formulas thatare given in the mentioned paper [16], depending on thechosen probability distribution of fuzzy events, whichmay be uniform or triangular (proportional) one.

,)~(

)~()~(

1

ij

ij

ijeijc F

c M c xe == .))~((,))~((

)~(

)~()~( 2/122

ijijijije

ij

ij

ijij cvc xc F

c M cv =−= σ (43)

Za uniformnu raspodelu slučajne rasplinutepromenljive:

For the uniform distribution of a random fuzzyvariable are:

,

3

)(2

2

)~( ul ul ij

C C B Bc F

−+

−= (44)

,5

)(2

3

)(2

4

)(3

32)~(

2222

1ul uul l uul l ul uul l

ij

C C C AC AC BC B B B B A B Ac M

−+

−+

−+

−+

−= (45)

+−

+−

+−

+−

=3

)(2

6

)(5

42)~(

22223322

2uul l uul l ul uul l

ij

B A B AC BC B A B B A B Ac M

+−

+−

+−

+3

)(2

5

)(4

2

)(3 2222

uul l uul l uuul l l C AC AC BC BC B AC B A (46)

5

)(4

7

)(2 2233

uul l ul C AC AC C −+

−+ .

Za trougaonu (proporcionalnu) raspodelu slučajnerasplinute promenljive su: For the triangular (proportional) distribution of arandom fuzzy variable are:

,2

)(2

3)~( ul ul

ij

C C B Bc F

−+

−= (47)

,325

)(3

43)~(

2222

1ul uul l uul l ul uul l

ij

C C C AC AC BC B B B B A B Ac M

−+

−+

−+

−+

−= (48)

+−

+−

+−

+−

=27

)(5

53)~(

22223322

2uul l uul l ul uul l

ij

B A B AC BC B A B B A B Ac M

+−

+−

+−

+

23

)(2

5

)(6 2222

uul l uul l uuul l l C AC AC BC BC B AC B A (49)

3

)(2

4

2233

uul l ul C AC AC C −+

−+ .

U ovim izrazima su: In these expressions are:

),)((,)()(, )()()()()()()()()()()()( l

ij

m

i

l

ij

m

ijl

l

ij

l

i

m

i

l

i

l

ij

m

ijl

l

i

l

ijl wwaaC awwwaa Bwa A −−=−+−== (50)

),)((,)()(, ))()()()()()()()()()()( u

ij

m

i

u

ij

m

iju

u

ij

u

i

m

i

u

i

u

ij

m

iju

u

i

u

iju wwaaC awwwaa Bwa A −−=−+−== (51)

.,...,2,1;,...,2,1 n jmi ==




56

5.3 Sračunavanje očekivanog idealno pozitivnog iidealno negativnog rešenja

Po kolonama matrice očekivanih vrednosti E, zasvaki kriterijum C j pronalazi se očekivano pozitivno

idealno rešenje*

je i negativno idealno rešenje− je po

sledećim formulama

5.3 Calculation of the expected ideal positive andideal negative solutions

For every criterion C j are found the best expected

positive ideal solution*

je and the worst negative ideal

solution− je

in the columns of the matrix of expected

values E by the following formulae

:or : minmax*

cijibijii je jee Ω∈Ω∈= , (52)

.:or :maxmin

cijibijii je jee Ω∈Ω∈=− (53)

Ove vrednosti su elementi vektora očekivanog

idealno pozitivnog (EPIS) *

e A i očekivanog idealno

negativnog rešenja (ENIS) −e A

These values are elements of vectors of the

expected positive ideal solution (EPIS) *

e A and expecte

negative ideal solution (ENIS) −e A solution

],...,,[ **

2

*

1

*

neee A = , ],...,,[ 21

−−−− = neee A (54)

Varijanse koje odgovaraju očekivanim vrednostimaoznačene su sa

−ii vv i

*i one čine vektore

Variances that correspond to these expected valuesare denoted as

*

iv and−iv and they constitute vectors

],,...,,[**

2

*

1

*

nvvvV = ].,...,,[ 21

−−−− = nvvvV

(55)

5.4 Određivanje očekivanog Euklidoveog rastojanjai njegove varijanse od EPIS i ENIS

Očekivana Euklidova rastojanja−ii ED ED i*

za

svaku alternativu Ai od očekivanog pozitivnog rešenja

EPIS *

e A i očekivanog negativnog idealnog rešenja

ENIS −e A određuju se prema sledećim formulama

5.4 Calculation of the expected Euclideandistances and its variance from EPIS and ENIS .

The expected Euclidean distances −ii ED ED i* for

every alternative Ai from the expected positive ideal

solution EPIS *

e A and from the expected negative idealsolution ENIS −

e A are calculated by formulas

;,...,2,1,])([ 2/1

1

2** miee EDn

j

jiji =−= ∑=

(56)

.,...,2,1,])([2/1

1

2miee ED

n

j

jiji =−= ∑=

−− (57)

Varijanse*

jV rastojanja alternative A j od

očekivanog pozitivnog idealnog rešenja EPIS *

e A i

varijanse−

jV od očekivanog negativnog idealnog

rešenja ENIS −e A , sračunavaju se prema sledećim

formulama, uzimajući u obzir pravilo o sabiranju ioduzimanju varijansi međusobno nezavisnih slučajnihvrednosti

Variance*

jV of the distance of alternative A j from

the expected positive ideal solution EPIS *

e A and

variance−

jV from the expected negative ideal solution

ENIS −e A , are calculated by the following formulas,

taking into account rule for summation and subtraction ofvariances for the mutually independent randomvariables

∑ ∑= =

−− =+=+=n

j

n

j

jiji jiji mivvV vvV 1 1

** .,...,2,1),(),( (58)






58

pozitivnog idealnog rešenja*

i ED >*

k ED i

**

k i CV CV < . Prema ovom pravilu rangiranja,

alternativa Ak je bolje rangirana od alternative Ai . slučaju,alternativu Ak treba bolje rangirati od alternative Ovajzaključak neće biti prihvaćen od strane donosioca

odluka ako je razlika između*

iCV i*

k CV mala. U tom

Ai , naročito kada alternativa Ak ima manju očekivanu

relativnu blizinu od alternative Ai , tj. .**

ik ERC ERC <Rangiranje prema očekivanoj relativnoj blizskosti imaprednost u odnosu na druga pravila rangiranja. Međutim,u praksi treba koristi sva pravila, a potom analiziratidobijene rezultate i donosiocu odluke predložitialternativu koja maksimalno zadovoljava ova pravila.

6 RASPODELA RASPOLOŽIVE KOLIČINE NOVCAZA ODRŽAVANJE OBJEKATA

Raspoloživa količina novca Q, opredeljena za

održavanje objekata, može se raspodeliti na osnovudobijene rang-liste, prema sledećim formulama− za rang-listu prema ERC i

*

**

k i CV CV < . According to this ranking rule, alternative

Ak is better ranked then alternative Ai . This conclusionwill not be accepted by the decision maker if differences

between*

iCV and*

k CV are small. In such a case

alternative Ak will be ranked better then alternative Ai ,especially when alternative Ak has smaller expected

relative closeness then alternative Ai , i.e.**

ik ERC ERC < . Ranking according to the expected

relative closeness have advantage over other rules.However, in practice all the rules should be applied,then, the obtained results analyzed and the alternativewhich best satisfies these rules should be proposed tothe decision maker.

6 DISTRIBUTION OF AVAILABLE AMMOUNT OFMONEY FOR OBJECTS MAINTENANCE

An available amount of money Q, which is assignedfor the maintenance of considered objects, should be

delivered according to the obtained rank list by thefollowing formulae

− for the rank list according to ERC i *,

Qci = (KIC )i Q, i =1,2,…m; (63)

− za rang listu prema CV i * − for the rank list according to CV i

* ,

Qvi = (KIV )iQ, i =1,2,…m; (64)

gde su (KIC )i i (KIV )i koeficijenti raspodele količine novcaQ, koji se sračunavaju prema sledećim formulama

where (KIC )i and (KIV )i coefficients of distribution of theamount of money Q that are calculated according to thefollowing formulas

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

∑∑*

*

*

*

mi

CV

CV KIV

ERC

ERC KIC m

i

i

i

im

i

i

i

i21===

1=1=

(65)

Prema izloženoj proceduri, autori su napisaliodgovarajući računarski program FUZZY_TOPSISkorišćenjem MATLAB programskog sistema.

7 PRIMER

Ovaj primer, koji je u vezi s procenom rizika mostova,preuzet je iz rada Wang i Ehlang [17],[18], u kome jeproblem rešen na sasvim drugačiji način.

Prema Britanskoj agenciji za auto-puteve [2], rizikmosta definiše se kao bilo koji događaj ili hazard koji

može onemogućiti postizanje poslovnih ciljeva iliostvarivanja očekivanja zainteresovanih strana (vlasnika,

deoničara, korisnika i dr.) i definiše se kao proizvodverovatnoće i posledice ostvarenog događaja.

U primeru je analizirano pet mostovskih konstrukcijaBS1,BS2,…,BS5 koje su predstavljene kao alternative A1, A2,…, A5. Sve posledice i verovatnoće rizičnihdogađaja procenjene su na osnovu evidencija i procenatri inženjera eksperta, imajući u vidu četiri kriterijuma:sigurnost (C 1), funkcionalnost (C 2), održivost (C 3) iokruženje (C 4). Eksperti su takođe procenili ikoeficijenate značaja alternativa. Ove vrednosti izraženesu kao lingvističke i numeričke vrednosti, koje su

According to this procedure, the authors have writtena corresponding computer program FUZZY_TOPSIS inMATLAB programming system.

7 EXAMPLE

This example, which is related to the bridge riskassessment, is taken from papers written by Wang andEhlang [17],[18] where this problem is solved in quitedifferent way.

According to British Highway Agency [2] bridge risk

is defined as any event or hazard that could hinder theachievement of business goals or the delivery ofstakeholder expectations and it is defined as product of the likelihood (probability) and consequence of theoccurred event.

In this example five bridge structures BS1, BS2,...,BS5 are considered which represent alternatives A1, A2,…, A5. All consequences and probabilities of the riskevents are assessed on the base of evidence andengineering judgment by three experts against fourcriteria: safety (C 1), functionality (C 2), sustainability (C 3)and environment (C 4). The significance coefficients ofalternatives are also assessed by experts. These values




59

konačno transformisane u trougaone fuzzy brojeve.Dobijene vrednosti su elementi fuzzy matrice odlučivanja

F~

= (Fl , Fm,,Fu) i predstavljaju nivo rizika konstrukcijemosta BSi u odnosu na kriterijum C j (i =1,2,..,5; j =1,2,..,4). Zadatak je odrediti optimalnu shemu(redosled rangiranja) i koeficijente raspodele količinenovčanih sredstava Q za održavanje mostova.

are assessed as linguistic and numeric variables that arefinally transformed into triangular fuzzy numbers. These

values are elements of the fuzzy decision matrix F~

=(Fl ,

Fm,,Fu) and denotes levels of risk of bridge structure BSi

against criterion C j (i =1,2,..,5; j =1,2,..,4). The task is todetermine optimal scheme (rank order) and coefficientsof distribution of available amount of money Q for thebridge maintenance.

Fl =

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

736200

2762620

15107327

22386262

15623873

, Fm =

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

858500

6285850

50388562

50738585

50857385

, Fu =

⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢

⎣

⎡

100100105

901001005

857310090

7895100100

8510095100

.

wl = [0.77 0.50 0.30 0.13], wm = [0.93 0.70 0.50 0.30], wu = [1.00 0.87 0.70 0.50] .

Pošto se rangira prema najvećem riziku, podskupovi

bΩ i cΩ jesu

Since the rank order is calculated according to high

level of risk, the subsetsΩ

b andΩ

c are

=Ω=Ω cb C C C C ),,,,( 4321 Ø.

Korišćenjem računarskog programa FUZZY TOPSIS,koji su razvili autori ovog rada, dobijeni su odgovarajućirezultati, sumirani u sledećoj tabeli.

The corresponding results summarized in thefollowing table are obtained by using computer programFUZZY TOPSIS developed by the authors of this paper.

Tabela 1. Sumarni rezutatiTable 1. Summary results

Rangalternative

Rank ofalternative

Očekivanaudaljenost. alter.EDi

* Expected distance of

altern. EDi *

Očekivana relat. blisk.altern .ERC i

* Expect. relat closen.

of altern. ERC i *

(KIC )i %

Koef. varijansealternative CV i

* Coeffic. of var. of

alternat. CV i *

(KIV )i %

1 A2=BS2 0.1203 A2=BS2 0.1142 28.7 A2=BS2 0.9089 28.62 A1=BS1 0.1402 A1=BS1 0.1322 28.1 A1=BS1 0.8455 26.53 A3=BS3 0.3141 A3=BS3 0.2846 23.1 A3=BS3 0.7510 23.54 A4=BS4 0.7684 A4=BS4 0.5651 14.1 A4=BS4 0.4795 15.05 A5=BS5 0.9584 A5=BS5 0.8129 6.0 A5=BS5 0.2028 6.4

Na osnovu dobijenih rezultata, datih u ovoj tabeli,može se zaključiti:

• Konstrukcija mosta BS2 (alternativa A2) imanajmanju očekivanu udaljenost od pozitivnog idealnogrešenja, tj. rešenja s najvećim nivoom rizika;

• Konstrukcija mosta BS1 (alternativa A1) ima svekarakteristične vrednosti koje su vrlo bliske vrednostimakonstrukcije BS2, pa tako ove dve konstrukcije imaju

praktično isti nivo rizika i zahtevaju iste količine novca zaodržavanje;• Konstrukcije mostova BS4 i BS5 imaju manje

karakteristične vrednosti i manji nivo rizika, pa samim timzahtevaju manju količinu novca za održavanje odkonstrukcija BS1 i BS2;

• Redosledi na osnovu očekivane relativne bliskostiERC i

* i generalizovanog koeficijenta varijacije CV i * u

ovom slučaju su isti;• Koeficijenti raspodele investicija (KIC )i i (KIV )i u

ovom slučaju su vrlo bliske vrednosti za sve konstrukcijemostova.

•

According to the obtained results, given in this table,the following may be concluded:

• Bridge structure BS2 (alternative A2) has thesmallest value of the expected distance from idealpositive solution, i.e. solution with the highest values ofdegree of risk;

• Bridge structure BS1 (alternative A1) has allcharacteristic values that are very close to BS2, so that

these two structures have practically the same degree ofrisk and require the same amount of money for themaintenance;

• Bridge structures BS4 and BS5 have smallercharacteristic values and smaller level of risk, so thatthey require smaller amount of money for themaintenance in comparison with structures BS1 and BS2;

• Rank list made by the expected relative closenessERC i

* and generalized coefficient of variation CV i * in this

case are the same;• Coefficients of investment distribution (KIC )i and

(KIV )i are very close for all bridge structures in this case.




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8 ZAKLJUČAK

Rasplinuti TOPSIS metod omogućava kompletnije,fleksibilnije i realnije modeliranje višekriterijumskogodlučivanja od nerasplinutog TOPSIS metoda s fiksnimvrednostima. U rasplinutom TOPSIS metodu moguće jeuvesti neprecizne ulazne podatke za matricu odlučivanjai težine kriterijuma. Metod predložen u ovom raduzasnovan je na generalisanoj očekivanoj vrednosti ivarijansi proizvoda elemenata matrice odlučivanja itežina kriterijuma. Za ove proizvode izvedene suodgovarajuće formule za njihovo tačno sračunavanje.Stoga, predloženi metod pruža donosiocu odluka tačnijei relevantnije rezultate nego klasični TOPSIS, što jevažno za donošenje korisnih odluka. Ovaj metod, uzkorišćenje pomenutog računarski programa, jeupotrebljen za rangiranje alternativa, odnosno varijantitrase za buduću železničku prugu Pljevlja – Bijelo Polje – granica sa Kosovom i Metohijom, kao I za još nekeinvesticione projekte. Metod može biti korisnoupotrebljen za rangiranje alternativa i optimalnuraspodelu investicionih sredstava na projekte, optimalnuprocenu rizika različitih tipova objekata, optimalan izbor

objekata za rekonstrukciju, izbor najpovoljnijeg izvođačaradova na tenderskim procedurama i u mnogim drugimslučajevima višekriterijumskog odlučivanja ugrađevinarstvu.

ZAHVALNOST

Ovaj rad je deo naučnoistraživačkog projekta"Upravljanje realizacijom velikih investicionih projekata"

TR ‒16011", koji je podržalo Ministarstvo prosvete i

nauke Republike Srbije i koji je urađen na Građevinskomfakultetu Univerziteta u Beogradu.

8 CONCLUSION

Fuzzy TOPSIS method enables more complete,flexible and realistic modelling of the multiple criteriadecision making problems then crisp TOPSIS methodwith crisp values. In the Fuzzy TOPSIS it is possible tointroduce imprecise input data for the decision matrixand weights of criteria. The method proposed in thispaper is based on the generalised expected values andvariances of products of the decision matrix elementsand weights of criteria. Thus, corresponding formulasand their exact calculation are derived for theseproducts. Therefore, proposed method provides moreaccurate and relevant results for the decision maker incomparison with classic TOPSIS, which is important foruseful decision making. This method with correspondingcomputer program is used for ranking traces of thefuture railway Pljevlja – Bijelo Polje – Border withKosovo and Metohija, as for some ather investmentprojects. The method may be used successfully forranking of alternatives and optimal distribution ofinvestments on the projects, optimal risk assessment ofdifferent types of objects, optimal choice of objects for

reconstruction, choice of the most acceptable contractorin tender procedures and in many other cases ofmulticriteria decision making in the Civil Engineering.

ACKLOWLEDGEMENT

This paper is the part of the scientific-researchproject "Menagement of realization of large investmentprojects TR-16011", supported by the Ministry ofeducation and science of the Republic of Serbia andrelised at the Faculty of Civil Engineering, University ofBelgrade.

9 LITERATURAREFERENCES

[1] Adey, B., Hajdin, R., Brühwiler, E.(2003), „Risk-based approach to the determination of optimalinterventions for bridges affected by multiplehazards”, Engineering Structures, Vol. 25, pp. 903-912.

[2] British Highway Agency (2004), Valuemanagement of the structures renewal programme,Version 2.2.

[3] Chen, S.-J. and Hwang, C.-L.,(1992),„Fuzzymultiple attribute decision making methods and

application”, Lecture Notes in Economics anMathematical Sciences, Springer, New York.[4] Cheng, C.-H. (1998), „A new approach for ranking

fuzzy numbers by distance method”. Fuzzy setsand systems, Vol. 95, pp. 307–317.

[5] Dubois, D. and Prade, H. (1978), „Operations onfuzzy numbers”, Int. Journal of System Sciences,Vol. 9, pp 307/317.

[6] Ertugrud, I. and Karakasogly, N. (2008), „Compa-rison of fuzzy AHP and TOPSIS methods for facilitylocation”, International Journal of AdvanceManufacture and Technology , Vo. 39, pp. 783–795.

[7] Hwang, C-L., Yoon, K. (1981), Multiple AttributeDecision Making an Applications, Springer-Verlag,New York.

[8] Folić, R., (editor) (2011), Proc. of VI Conf.„Assessment, Maintenance and Rehabilitation ofStructures and Settlements”, Association of CivilEngineers of Serbia, 710 p.

[9] Folić, R., (editor) (2011), Proc. of VII Conf.„Assessment, Maintenance and Rehabilitation ofStructures and Settlements”, Association of Civil

Engineers of Serbia, 608 p.[10] Folić, R., (editor) (2013), Proc. of VIII Conf.„Assessment, Maintenance and Rehabilitation ofStructures and Settlements”, Association of CivilEngineers of Serbia, 638 p.

[11] Lee, E. S and Li, R. L. (1988), „Comparison offuzzy numbers based on probability measure offuzzy events”. Comput Math Appl , Vol. 15, pp.887–896.




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[12] Opricović, S (1998), Muliple criteria optimization of systems in Civil Engineering (in Serbian), Univer-sity of Belgrade, Faculty of Civil Engineering,Belgrade.

[13] Opricović, S., Tzeng, G-H. (2004), „Compromisesolution by MCDM methods: A comparative

analysis of VICOR and TOPSIS”, Europ. J. of Operational Research, Vol. 156, pp. 445–455.

[14] Opricović, S.(2007), „A Fuzzy CompromiseSolution for Multicriteria Problems”, Int. J. oUncertainty, Fuzziness and Knowledge-baseSystems, Vol. 15, No. 3, pp. 363–380.

[15] Praščević, Ž., Praščević, N.,(2011), „Application of Fuzzy TOPSIS Method for Choice of Objects for Reconstruction and Maintenance”, Proc. of VIIConf. „ Assessment, Maintenance and Rehabili-tation of Structures and Settlements” , edited by R.Folić, Association of Civil Engineers of Serbia, pp25–33. Prascevic, Ž., Praščević, N. (2013), „One

modification of fuzzy TOPSIS method”, Journal oModelling in Management , Vol. 8, pp.81–102.

[16] Wang, Y-M. and Elhang, T. M. S. (2006), “FuzzyTOPSIS method based on alfa level with anapplication to risk management”, Expert Systemswith Application Vol. 31, pp. 309-319.

[17] Wang, Y-M. and Elhang, T. M. S. (2007), „A fuzzygroup decision making approach for bridge riskassessment”, Computers and Industrial Engine-ering , Vol. 53, pp. 137–148.

[18] Zadeh, L. A. „Fuzzy Sets”(1965), Information andControl, Vol. 8, pp. 338-353.

[19] Zadeh, L. A.,(1975), „Probability Measure of FuzzyEvents”, Journal of Mathematical Analysis and its Applications, Vol. 23, No 2, pp. 421–428.

[20] Zimmermann, H-J., „Description and optimizationof fuzzy systems”, Intern. Journal of GeneralSystems, 1976, Vol. 2, pp. 209–215.

REZIME

PRIMENA MODIFIKOVANOG RASPLINUTOG TOPSIS METODA ZA VIŠEKRITERIJUMSKE ODLUKE UGRAĐEVINARSTVU

Živojin PRAŠČ EVI Ć Nataša PRAŠČ EVI Ć

U ovom radu predlaže se i primenjuje jedanmodifikovani rasplinuti TOPSIS metod za višekriterijum-sko rangiranje objekata za rekonstrukciju i održavanje.Na početku se daje kratak osvrt na nastanak i razvojovog metoda i opisuje se TOPSIS procedura s fiksnim

(nerasplinutim) ulaznim podacima koji sačinjavaju matri-cu odlučivanja i težinske koeficijente kriterijuma. Ovaprocedura se ilustruje jednim jednostavnim brojčanimprimerom. Objašnjava se neophodnost prikazivanja ovihparametara - kao trougaonih rasplinutih brojeva - zbognemogućnosti njihovog preciznog određivanja ili proce-njivanja u praksi. U radu se daju tačni izrazi, koje suautori ranije izveli, za određivanje proizvoda elemenatamatrice odlučivanja i težinskih koeficijenata kao trougao-nih rasplinutih brojeva. Ovi parametri za svaku alterna-tivu (objekat) tretiraju se kao slučajne rasplinute veličine,za koje se određuju tačne generalisane očekivanevrednosti, varijanse i standardne devijacije. Iz normalizo-vane matrice očekivanih vrednosti određuju se očekiva-na idealna pozitivna i očekivana idalna negativna reše-nja. Za svaku alternativu određuju se generalisane oče-kivane distance i relativne bliskosti ovim rešenjima, kao iodgovarajuće varijanse i koeficijenti varijacije. Alternativese rangiraju prema ovim vrednostima. U radu sepredlažu izrazi za sračunavanje koeficijenta raspodeleinvesticionih sredstava na (alternative) objekte. Na kraju,dat je jedan primer rangiranja mostovskih konstrukcija uodnosu na rizik i formulisani su odgovarajući zaključci.

Ključne reči: Rasplinuti (fuzzy) TOPSIS, rasplinutibroj, održavanje objekata, raspodela investicionihsredstava,upravljanje rizikom.

SUMMARY

APPLICATION OF MODIFIED FUZZY TOPSISMETHOD FOR MULTICRITERIA DECISIONS IN CIVILENGINEERING

Živojin PRASCEVICNataša PRASCEVIC

In this paper is presented and applied one fuzzyTOPSIS method for the multicriteria rancing of objectsfor reconstruction and maintenance. At the beginning isgiven short review on the genesis and development ofthis method and described a TOPSIS procedure with

crisp input data that constitute a decision matrix andweights of criteria. This procedure is ilustrated by onesimple numerical example. The necessity of presentationof these parameters as triangular fuzzy numbers due toimpossibility of their precise determination orassessment in the practice. The exact expressions forthe determination of these products of the decisionmatrix and weights coefficients as triangular fuzzynumbers, that authors of this paper are derived earlier,are given in the paper. For every alternative (the object)these parameters are assumed as random fuzzynumbers for which are determined generalised expectedvalues, variances and standard deviations. From thenormalised matrix of the expected values are determinedexpected ideal positive and ideal negative values. Forevery alternative are determened generalized expecteddistances and relative closenesses to the ideal positiveand ideal negative solution. The ranking of alternatives isperformed according to these values. Matematicalexpressions for coefficients of investments distributionon the alternatives (objects) are proposed in the work.One example of ranking of the bridge structuresaccording to the risk is given at the end of the work andformulated corresponding conclusions.

Key words: Fuzzy TOPSIS, fuzzy number,maintenance of objects, distribution of investments, riskmanagement.




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63

KARAKTERISTIKE TRANZITNE ZONE BETONA NA BAZI LAKOG AGREGATA

CHARACTERISTICS OF LIGHTWEIGHT AGGREGATE CONCRETE INTERFACIALTRANSITION ZONE

Vesna MILOVANOVI Ć

Vilma DUCMANMiroslava RADEKA

ORIGINALNI NAUČNI RADORIGINAL SCIENTIFIC PAPER

UDK: 691.322

1 UVOD

Proizvodnja konstrukcijskih betona s poboljšanimtermoizolacionim svojstvima na bazi kvalitetnog lakogagregata (LA) proizvedenog od otpadnih materijalapredstavlja novu, dodatnu vrednost koja utiče naracionalno korišćenje prirodnih resursa i upravljanjeotpadom. S lakim agregatom mogu se uspešnoproizvoditi betoni za monolitne konstrukcije kao iprefabrikovani elementi. Dobre strane upotrebe lakihagregata proizvedenih na ovaj način jesu:

• zamenjuju prirodne sirovine zbog čega se smanjujekoličina otpadnog materijala na deponijama;

• povećavaju uštede u izgradnji i eksploatacijiobjekata smanjenjem opterećenja na temelje (do 25%),čime se povećava otpornost konstrukcije na dejstvozemljotresa;

• povećavaju toplotno-izolaciona svojstva elementakonstrukcije.

Kao polazna sirovina za proizvodnju ove vrsteagregata najčešće se koristi: zgura visokih peći, letećipepeo, otpadno staklo, silikatni mulj, papirni mulj, itd. [4],koji po svojim karakteristikama uspešno konkurišu lakimagregatima dobijenim od prirodnih sirovina: vermikulit,perlit i glina.

Vesna Bulatović, asistent, Univerzitet u Novom Sadu,Fakultet tehničkih nauka - Departman za građevinarstvo igeodeziju, Trg Dositeja Obradovića 6, Novi Sade-mail: [email protected] Ducman, Zavod za gradbenistvo, 11001 Ljubljana,Dimičeva 12, Slovenia, e-mail: [email protected] Radeka, profesor, Univerzitet u Novom Sadu,Fakultet tehničkih nauka - Departman za građevinarstvo igeodeziju, Trg Dositeja Obradovića 6, Novi Sade-mail: [email protected]

1 INTRODUCTION

The production of construction concrete withenhanced thermal insulating characteristics based onquality lightweight aggregate (LWA) produced fromwaste materials, represents a new, additional qualitywhich influences rational usage of natural resources andwaste management [3,4]. Concrete for monolithconstructions along with prefabricated elements can beeasily produced with lightweight aggregate. Positivesides of the use of lightweight aggregate produced in

this way are:• Natural materials are replaced in this way, which

results in reduced amount of waste material deposited inlandfills

• There is a reduction in building expenses byreduction of deadweight of the structure (up to 25%) andincrease of overall earthquake resistance

• Thermal insulating properties of constructionelements are increased

Mostly used raw materials for production of LWA are:blast furnace slag, fly ash, waste glass, silicate sludge,paper sludge [4] etc., all of which successfully canreplace LWA made from natural materials: vermiculit,perlite and clay.

VesnaBulatović, assistent, University in Novi Sad, Facultyof Technical sciences-Department of Civil Engineering andGeodesy, TrgDositejaObradovica 6, Novi Sad,e-mail: [email protected], Zavodzagradbenistvo, 11001 Ljubljana,Dimičeva 12, Slovenia, e-mail: [email protected], professor, University in Novi Sad, Facultyof Technical sciences-Department of Civil Engineering andGeodesy, TrgDositejaObradovica 6, Novi Sade-mail: [email protected]




64

Beton na bazi lakog agregata ima u svežem iočvrslom stanju osobine drugačije od običnog betona.One su posledica osobina lakog agregata (malazapreminska masa, dobre termoizolacionekarakteristike, sposobnost da upijenu vodu koristi zanegu betona u procesu očvršćavanja) [7] po kojim serazlikuje od osobina normalnog agregata. Osobineočvrslog betona na makronivou, na kom se definišu

inženjerska svojstva betona, posmatraju se kao osobinehomogenog materijala, dok se na mezonivou smatra dase sastoji od tri faze: 1) agregat 2) cementna pasta i 3)tranzitna zona (TZ) između ove dve faze. Uticaj TZ načvrstoću i propustljivost betona veoma je velika.Teksturalna i mineraloška svojstva površine lakogagregata utiču na formiranje mikrostrukture u ovoj zoni.Iz ove činjenice je proistekao značaj proučavanjamikrostrukture formirane tranzitne zone, jer jeprepoznato da interakcija zrna agregata s cementnompastom u velikoj meri modifikuje osobine betona upogledu čvrstoća, modula elastičnosti i propustljivosti zagasove i tečnosti [2]. Tranzitna zona koja se formiraizmeđu agregata normalne težine i očvrslog cementnogkamena je porozna (u betonima normalne težine). Većaporoznost je pripisana „efektu zida”, koji vodi kasmanjenju sadržaja zrna nehidratisanog cementa upoređenju s cementnom matricom, usled čega dolazi dorasta poroznosti [12a]. To znači da za isti vodocementnifaktor u tranzitnoj zoni ima manje nehidratisanih zrnanego u cementnoj matrici. Ako je vodocementni faktor uceloj masi 0.4, u cementnoj matrici će biti 0.36 [12b].

Prema literaturnim podacima [10], formiranjetranzitne zone se odvija u nekoliko faza:

• prvo se formira film od vode na površini zrnaagregata što dovodi do povećanja vodocementnogfaktora;

• kasnije, kao i u cementnoj pasti, dolazi dorastvaranja kalcijum-sulfata i kalcijum-aluminata. Najvećideo zrna cementa sastavljan je od više minerala i napočetku preovlađuju reakcije minerala C3 A i C3S. Uprvim momentima hemijske reakcije stvara se amorfnisloj bogat aluminijumom. Iz ovog sloja počinju da serazvijaju kristali etringita (AFt), a kada počne hidratacijatri kalcijum-silikata (C3S), na kristalima AFt počinje rastkalcijum-siliko-hidrata (CSH) i popunjavanje praznogprostora. Na kraju ove etape formiraju se pore sprečnikom manjim od 1µm. Hidratacija se potomnastavlja rekristalizacijom AFt i formiranjem većihkristala. Uvećanje ovih kristala i kontinuiran rast CSHprodukata na kraju dovodi do bolje popunjenosti prostoračvrstom materijom i smanjenjem poroznosti čime sepoboljšava gustina i čvrstoća TZ [10];

• zbog većeg vodocementnog faktora u tranzitnojzoni u odnosu na cementnu matricu, joni formiraju većekristale i porozniju strukturu.

Proces hidratacije se takođe menja i pod uticajemsuperplastifikatora. Njihova uloga je da menjaju reološkasvojstva svežeg betona [11] u interakciji, pre svega saC3 A i njegovim produktima hidratacije. Oni takođeusporavaju proces hidratacije alita, a s povećanjem

sadržaja C3 A, količina polimera koji se adsorbuje je veća

[15].

Concrete based on LWA (LWAC) in its fresh and hardcondition has qualities different from ordinary concrete.They come as the consequence of LWA properties (smallapparent density, good thermo insulating properties andthe ability to use absorbed water for internal curing duringhardening process) [7] which makes it different fromordinary concrete. The characteristics of hardenedconcrete on macro level, which is used to define

engineering characteristics of concrete, are seen as thecharacteristics of homogenous material, whereas it isconsidered that at mezo level there are three phases: 1)aggregate 2) cement paste (matrix) and 3) an interfacialtransition zone (ITZ) which is between these two phases.The influence of ITZ to the strength and permeability ofconcrete is very high. Textual and mineralogicalcharacteristics of the surface of LWA influence theformation of microstructure in this zone. Thus, it isimportant to study microstructure of the formed ITZ sinceit has been recognised that the interaction of aggregatewith cement paste to a great extent modifies thecharacteristics of concrete when it comes to hardness, themodule of elasticity and permeability to gases and liquids[2]. ITZ formed between the normal aggregate andhardened cement paste is porous (in ordinary concrete).The higher porosity is due to a "wall effect", leading tolower content of unhydrated cement grains as comparedto bulk phase and consequently to an increased porosity[12a]. This means that the volume fraction of unhydratedcement grains is lower in ITZ than in cement matrix for thesame water cement ratio. The reduction of water cementratio in the bulk phase was expected to be about 0.36compared to overall w/c of 0.4 [12b].

According to the data given in the reference [10] theformation of ITZ takes place through several stages:

• firstly, a water film is being formed on the surfaceof aggregate grain which leads to the increase of thewater/cement ratio in the vicinity of aggregate grain;

• then, as it happens in the cement paste, in the ITZoccurs dissolving of calcium sulphate and calciumaluminate. Most cement grains are multi mineral andearly reactions are dominated by C3 A and C3S reactions.In the first moments of a chemical reaction there comesto the formation of an amorphous layer rich inaluminium. From this layer start the formation ofettringite crystals (AFt), and when the hydration C3Sstarts, on crystals AFt start to grow CSH compounds andto fill porous space. At the end of this stage the diameterof the pores that are being formed is less than 1µm.Hydration then continues with recrystallization of AFt andthe creation of bigger crystals. The enlargement of thesecrystals and the continuous growth of CSH compounds

eventually bring to a better filling of space with productsof hydration and decreased porosity which improvedensity and strength of ITZ [10];

• due to the higher water/cement ratio in ITZ than incement matrix, the ions form large crystals and thereforeporous structure appears.

The hydration process is also changing in thepresence of superplastisizer. They change the rheology[11] of the fresh concrete by adsorbing preferably on theC3 A phase. They also adsorb on C3 A hydration productsand retard alite hydration. With the increasing C3 Acontent of the cement the amount of polymer adsorbedis increasing. [15].






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2 MATERIJALI I METODE

Za spravljanje konstrukcijskog betona korišćene sudve vrste lakog agregata: LA na bazi otpadnih materija(Termit a.d., iz Slovenije) i komercijalni laki agregatLeca-Laterlite (Italy). Najveći udeo u odnosu na ukupnukoličinu lakog agregata imao je laki agregat napravljenod silikatnog mulja, mulja iz papirne industrije i pepela[4]. Silikatni mulj je nastao prosejavanjem peska naiskopu. Pre pečenja sirovinska masa je granulisana.Temperatura pečenja je 900oC.

Upijanje vode i zapreminska masa samog zrna lakogagregata određene su na osnovu procedure date ustandard SRPS EN 1097-6:2007.

Čvrstoće zrna LA određene su prema proceduri datoju literaturi [8].

Na bazi ovog agregata i peska napravljene su četirivrste betona sa sledećim komponentnim materijalima:

Cement : CEM I 42,5R i CEM II/B-M(S-V-L)32,5R(Lafarge-BFC, Srbija), Leteć i pepeo (Termoelektrana„Nikola Tesla B” Obrenovac), Metakaolin („Metamax” –SAD)

Agregat : Sitan: prirodni rečni pesak (0/4 mm) i laki

agregat, Leca (2/3 mm), Italija; Krupan: laki agregat (4/8mm), proizvođač Termit d.d, Slovenija

Hemijski dodatak : Superplastifikator Sika Viscocrete4000BP

Voda: Gradski vodovodSastav betonskih mešavina, zajedno sa oznakama,

dat je u Tabeli 1.

2 MATERIALS AND METHODS

Two types of LWAs were used for the preparation ofstructural concretes: LWA produced from wastematerials and commercial one Leca-Laterlite (Italy). Themain part of total amount of LWAs was prepared of silicasludge, paper mud and ash [4]. The silica sludgeremains after sand screening in quarry (Termit d.d.,factory from Slovenia). Before firing the batch wasgranulated. Firing temperature was 900oC.

Water absorption and apparent density weredetermined according to the procedure given in standardSRPS EN 1097-6:2007.

Mechanical strength of single LWAs granules wasdetermined according to the procedure defined in [8].

Based on this LWAs and river sand, four types ofstructural concretes with the following consistingmaterials were made:

• Cement : CEM I 42,5R and CEM II/B-M (S-V-L)32,5R (Lafarge-BFC, Serbia), Fly ashe (Power plant"Nikola Tesla B" Obrenovac), Metakaolin ("Metamax"-SAD)

• Aggregate: small fractions: river sand (0/4mm) and

lightweight aggregates:, Leca (2/3mm), Italy; largefractions : lightweight aggregate (4/8mm), produced byTermit d.d, Slovenia

• Admixture: Superplastificator Sika Viscocrete4000BP

• Water : tap waterThe composition of structural concrete mixtures is

given in Table 1.

Tabela 1. Sastav betonskih mešavina, komponente su u kg/m3 Table 1. Composition of structural concrete mixtures, kg/m3

Vrsta betonaType of concrete

LSLL-1 LSLL-2 LSLL-4 LSLL-5

CEM I 42,5R 450 - 400 244CEM II/B-M(S-V-L)32,5R - 450 - -Voda / Water 180 180 180 183Dodatna voda / Additional water 119,5 119,5 121,2 119,5Rečni agregat / River aggregate (0/4mm) 749 749 771 760Leca-Laterlite / LWA Leca-Laterlite (2/3mm) 90 90 91 90Lakoagregatni agregat - Termit / LWATermit (4/8mm) 327 327 332 327Leteći pepeo / Fly ash - - - 97,6Metakaolin / Metakaolin - - - 24,4Sika VSC 4000BP / Sika VSC 4000BP 2,25 3,15 2,0 2,56

Da bi se sprečila absorbcija vode u većoj meri izcementne paste, što smanjuje količinu vode koja jeneophodna za hidrataciju cementa, LA se pre spravljanjabetonske mešavine natapa vodom u trajanju od 30 min(dodatna voda). Količina upijene vode osigurava da LAbude „površinski zasićen” vodom.

Ukupna poroznost i raspodela veličina pora u lakomagregatu Termit proizvedenom u Sloveniji (LWA-Termit),određena je živinom porozimetrijom (MIP). Male kocke,veličine oko 1cm3, sušene su u sušnici 24 h na 105oC ianalizirane u uređaju Micromeritics Autopore IV 9500model. Uzorci su analizirani u opsegu pritisaka od 0 do414 MPa.

Karakteristike tranzitne zone određene su putem

In order to prevent great absorption of water from thecement paste, which decreases the water amountneeded for the cement hydration, LWAs were being pre-wetted with water for 30 minutes (additional water). Thisamount of absorbed water ensures ”surface-saturated”condition prior to mixing.

Total porosity and pore size distribution of lightweightaggregate Termit, produced in Slovenia (LWA-Termit),was determined by means of mercury intrusion porosity(MIP). Small block, approximately 1cm3 in size, weredried in an oven for 24h at 105oC and analysed on aMicromeritics Autopore IV 9500 model. Samples wereanalysed in the range of 0 to 414 MPa.

The characteristics of ITZ were determined by SEM






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LA – Termit, slika1, ima veoma veliku otvorenuporoznost, Tabela 2. Raspodela veličina pora, slika 2,ukazuje na to da je dominantan prečnik pora oko 1µm,dok je ukupna poroznost 46.5 %. Ova veličina poraomogućava dodatnoj vodi da iz ovih i većih pora curi uTZ i učestvuje u procesu hidratacije cementne paste. Naosnovu vrednosti ukupne i otvorene poroznosti utvr đeno je da je zatvorena poroznost oko 12%.

LWA-Termit, Figure 1, has a very high open porosity,Table 2. Pore size distribution, Figure 2, indicates thatdominant pore diameter is 1 µm, while total porosity is46.5%. This size of the pores enables additional waterfrom these and bigger pores to take part in the processof hydration of cement paste. Based on the values oftotal and open porosity it was determined that the closedporosity is around 12%.

Slika 1. Laki agregat-TermitFigure 1. Llightweight aggregate-Termit

Slika 2. Raspodela veli č ina pora u LA-TermitFigure 2. Pore size distribution in LWA-Termit

Slika 3. Rendgeno-strukturna analiza LA-Termit (Q- kvarc, G-gelenit)Figure 3. XRD analysis of LWA-Termit (Q- quartz, G-gehlenite)

Mineraloški sastav LA-Termit (slika 3) ukazuje na

prisustvo dva dobro iskristalisana minerala: gelenit ikvarc.

Na slici 4a predstavljeno je zrno LWA-Termit ubetonu LSLL1, starosti 14 dana. Zapaža se da ne postojioštra granica između zrna LA-Termit i cementne paste.Mikrostruktura poprečnog preseka LA-Termit, slika 4a,ukazuje na prisustvo šupljina i pukotina nastalih u tokuprocesa proizvodnje. Odsustvo guste spoljašnje opne,koja kontroliše absorpciju/desorpciju vode, trebalo bi daomogući dotok vode u cementnu pastu tokom hidratacijeu slučaju desorpcije. Na ovaj način, slojevi cementnepaste bliži površini LA postaju gušći za kraće vreme. Uunutrašnjosti zrna LWA mogu se videti kristalne forme

Mineralogical composition of LWA-Termit, (Figure 3)

shows the presence of two well crystallized minerals:gehlenite and quartz.Figure 4a shows a grain of LWA-Termit in concrete

LSLL1, 14 days old. It is evident that there is no distinctdifference between the LWA grain and the cementpaste. The microstructure of the LWA-Termit grain crosssection, Figure 4a, indicates the presence of voids andfissures formed during the manufacturing process. Thelack of outer shell, which controls absorption/ desorptionof water, should facilitate supply of cement paste withwater in the case of desorption. In that way near-surfacelayer of the cement paste becomes denser in a shorterperiod of time. Inside of the LWA grain nano sized




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nanoveličine, slika 4b, koje podsećaju na etringit čijaidentifikacija u ovoj veličini nije laka. Nastale sunajverovatnije nakon absorbcije vode u LWA, ali ovupojavu treba dodatno istražiti.

crystal forms can be seen, Figure 4b. They remind of thestructure of the ettringite (AFt) whose identification isdifficult due to a size effect. The crystal forms wereprobably made after the absorption of water in LWA.However, this occurrence needs further investigation.

Slika 4a.SEM LA-Termit u betonu LSLL1, starost 14dana (x200)

LWA-lak agregat, CP- cementna pastaFigure 4a. SEM of LWA-Termit in LSLL 1 concrete,14

days old (x200),LWA-lightweight aggregate, CP-cement paste

Slika 4b.SEM LA-Termit u betonu LSLL1, starost 14 dana(x200)

Figure 4b. SEM of LWA-Termitin LSLL 1 concrete,14 daysold, (x2000)

Iako na poprečnom preseku LA-Termit nije utvr đenoda postoji gusta spoljašnja opna, razlike u sastavupovršine i unutrašnjeg dela zrna jesu moguće. Porastvaranju neizreagovanih zrna cementa, javlja se

koncentracioni gradijent koji može dovesti do difuzije jona iz cementne paste do TZ i unutrašnjih delova LA.Najpokretljiviji joni u portland cementu su Na+, K+, Al(OH)4, Ca2+ [13].

Na osnovu EDS ispitivanja sadržaja elemenata utranzitnoj zoni u opsegu od -10 µm (počinje u zrnu lakogagregata) do ≈ 90 µm (završava se u cementnoj pasti), ikriterijuma datih u radu [14], određuje se odnos atoma(Ca, Si, Al, Fe, S) i tip hidradisanog produkta premasledećim formulama:

Although the shell structure is not seen in the crosssection of LWA grain the differences in oxidecomposition of surface and inner part of the LWA arepossible. After the dissolution of cement anhydrous

grains, the appearance of concentrations gradient couldinduce a transport of ions by diffusion from cement pasteto ITZ and inner part of LWA. The most mobile ions areNa+, K+, Al(OH)4

¯¯ , Ca2+ in ordinary portland cement[13].

EDS examinations of element composition in ITZranged between -10 µm (starts in the grain of LWA) to ≈ 90 µm (finished in cement paste), and the criteria givenin paper [14] determines the ratio of atoms(Ca,Si,Al,Fe,S) and type of hydrated products accordingto the following formulas:

C-S-H: 0.8≤Ca/Si≤2.5, (Al+Fe)/Ca≤0.2

CH: Ca/Si≥10, (Al+Fe)/Ca≤0.4, S/Ca≤0.04

AFm1: Ca/Si≥4.0, (Al+Fe)/Ca>0.4, S/Ca>0.15Na osnovu datih kriterijuma utvr đuje se koja

jedinjenja preovlađuju u hidratnim formama (CSH, CH, AFm). Ako rezultati pokažu sastav koji je između ovihgranica, smatra se da su formirane hidratne formemešavina oksida. Ideja je bila da se napravi razlikaizmeđu hidratnih oblika bogatih C-S-H jedinjenjima, jedinjenjima kalcijum-hidroksida (CH) i monosulfata(AFm) ili etringita (AFt). Primer proračuna je dat utabelama 3-7 za različite betone starosti sedam dana.

Based on the given criteria, it is determined whichcompounds are dominant in hydration forms (CSH, CH, AFm). If the results show structure which is betweenthese values then it is considered that the producedhydration forms are mixtures of oxides. The idea was tomake difference between hydrates rich in C-S-Hcompounds, compositions reach in calcium hydroxide(CH) and monosulphate (AFm) or etringite (AF t). Theexamples of calculations are given in Tables 3-7.

1 AFm su u hidratisanom cementu produkti hidratacije C3 A(monosulfat i monokarbonat).

1 Afm phases in hydrated cements are products of C3 Ahydration (monosulfate and monocarbonate)




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Na osnovu datih kriterijuma utvr đuje se koja jedinjenja preovlađuju u hidratnim formama (CSH, CH, AFm). Ako rezultati pokažu sastav koji je između ovihgranica, smatra se da su formirane hidratne formemešavina oksida. Ideja je bila da se napravi razlikaizmeđu hidratnih oblika bogatih C-S-H jedinjenjima, jedinjenjima kalcijum-hidroksida (CH) i monosulfata

(AFm) ili etringita (AFt). Primer proračuna je dat utabelama 3-7 za različite betone starosti sedam dana. U

Tabeli 3 je predstavljen hemijski sastav spoljašnje opnezrna LA-Termit. pre ugrađivanje u beton. Struktura LA-Termit u blizini TZ prikazuju tačke 1-3 u Tabeli 4, tačke 3i 4 u Tabeli 5 i 6, tačke 4-6 u Tabeli 7. Ostale tačkepokazuju sastav tranzitne zone.

Based on the given criteria, it is determined whichcompounds are dominant in hydration forms (CSH, CH, AFm). If the results show structure which is betweenthese values then it is considered that the producedhydration forms are mixtures of oxides. The idea was tomake difference between hydrates rich in C-S-Hcompounds, compositions reach in calcium hydroxide

(CH) and monosulphate (AFm) or etringite (AF t). Theexamples of calculations are given in Tables 3-7.Chemical composition of the LWA-Termit grain outershell before concrete preparation is presented in Table3; in Table 4 points 1-3 show structure of the grain ofLWA-Termit, in Tables 5, 6 these points are 3 and 4while in Table 7 are 4-6. The rest of the points show thecomposition of ITZ.

Tabela 3. Rezultati EDS analiza spoljašnje opne zrna LA-Termit, mas%Table 3. The results of EDS analysis of LWA -Termit grain outer shell, mass%

Spec. C O Mg Al Si K Ca Ti Fe Ca/Si (Al+Fe)/Ca S/Ca

Range24.6-45

47.9-50

0.57-0.86

4.43-12.9

0-180-0.4

11.2-30.2

0-0.3

0.46-0.74

Aver. 23.2 32.6 0.68 8.45 9.4 0.2 21.2 0.1 0.61 2.25 0.43 - mix

Analizom sastava tačaka 1 i 2 (spektri 1 i 2), Tabela4, koje se nalaze u zrnu LA-Termit vidi se povećanjekoncentracije kalijuma i gvožđa u odnosu na vrednosti uTabeli 3. Odnos Ca/Si raste od tačke 1 do 3. Rastodnosa atoma Ca/Si upućuje na postojanje difuzije jona.Zbog velikih razlika u sastavu zrna LA-Termit, Tabela 3,teško je reći u kom smeru se dominantno odvija difuzija(od cementne paste ka zrnu LA ili obrnuto, ili u obapravca).

By analyzing the composition of points 1 and 2(spectrum 1 and 2), Table 4, which are in the LWA-Termit grain, a certain increase of the concentration ofpotassium and iron can be seen in comparison to valuesin Table 3. The increase of Ca/Si ratio indicates thediffusion of ions between cementous paste and LWA-Termit grain. Due to the large differences in thecomposition of LWA-Termit grain, Table 3, it is difficult tosay in which direction the diffusion dominantly takesplace (from cement paste to the LWA grain or viceversa, or that it maybe goes in both directions) [16b].

Slika 5. Mesta na kojim je urađ ena EDS analiza 7 dana starog betona LSLL-1Figure 5. Spots on which EDS analysis was carried out for 7-day old concrete LSLL-1




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Tabela 4. Rezultati EDS analize betona LSLL-1 starog 7 danaTable 4. The results of EDS analysis of 7-day old concrete LSLL-1

Spec C O Mg Al Si K Ca Ti Fe Ca/Si Al+Fe/Ca

Tačke u LA-TermitPoints in LWA-Termit

Spec 1 0.00 49.40 0.87 6.17 15.29 1.46 25.61 1.21 1.67 0.29 mix

Spec 2 13.81 43.07 0.78 5.44 9.89 0.64 25.22 1.15 2.55 0.26 mixSpec 3 0.00 50.29 1.37 6.38 9.29 0.57 30.66 0.25 1.20 3.00 0.25 mix

Tačke u TZPoints in ITZ

Spec 4 25.39 42.75 0.49 3.29 5.43 0.84 21.27 0.55 3.91 0.18 mixSpec 5 16.78 47.54 0.77 2.43 5.07 1.20 25.92 0.29 5.11 0.10 mixSpec 6 13.39 44.52 0.89 4.77 7.69 1.22 26.66 0.86 3.47 0.21 mixSpec 7 14.65 45.36 0.43 4.46 13.79 0.79 19.79 0.73 1.44 0.26 mix

U betonu gde je kao vezivo korišćen CEM I 42,5R(LSLL1), na mestima 1 i 2, koja se nalaze u zrnu LWA-Termit, odnosi Ca/Si i (Al+Fe)/Ca bliski su vrednostima

za CSH jedinjenja, Tabela 4. Moguće je da zidovi poraLWA služe kao termodinamički najpovoljnija mesta zaformiranje CSH (heterogena nukleacija). U tačkama gdetreba da se nalazi TZ (tačke 4-7, Tabela 4) dobijeno jepovećanje vrednosti Ca/Si (tačke 4 i 5), a potomsmanjenje (tačke 6 i 7). Ovi rezultati su u skladu spretpostavljenom strukturom sloja TZ najbližeg zrnuagregata, jer ukazuju na povećanje koncentracijeportlandita. Naime ovaj deo ITZ trebalo bi da bude uvećoj meri sastavljen od kristala portlandita, a potom i odCSH, AFm i Aft jedinjenja.

Poređenjem vrednosti udela elemenata, pre svegaCa i Si u betonu LSLL1, sa betonom napravljenim odmanje količine istog cementa i većim vodocementnimfaktorom (LSLL4-CEM I 42,5R), Tabele 4 i 6, uočava sepovećanje koncentracije pomenutih, ali i drugih eleme-nata (kao Al i Fe) kod betona LSLL 4 (tacke 1 i 2- TZzona). Prema navodima datim u literaturi [12b] vodo-cementni faktor ima mali uticaj na mikrostrukturu tran-zitne zone. U uslovima povećanog vodocementnog fak-tora odvija se kontinualno rastvaranje i interakcija vodesa CSH. Posebno su visoke koncentracije elementa Si,

In concrete where CEM I 42,5R (LSLL1) was usedas a binder, in points 1 and 2 (Table 4 spectrum 1 and2), which are in LWA-Termit grain, the ratios Ca/Si and

(Al+Fe)/Ca are close to the values for CSH compounds.It is possible that the walls of the LWA pores are used asthermodynamically most favourable places for theformation of CSH (heterogeneous nucleation). In pointswhere there should be ITZ (points 4 - 7), there was anincrease in value of Ca/Si (points 4 and 5), and then adecrease (points 6 and 7). These results are inaccordance with the assumed structure of the layerclosest to the grain of the LWA-Termit indicating theincrease of mineral portlandit. Namely, this part of ITZshould be mostly composed of the portlandit crystals andCSH, AFm and Aft phases as well.

When comparing the values of the ratio of elements,first of all Ca and Si in concrete LSLL1, with concretemade from a less amount of cement and greater watercement ratio (LSLL4-CEM I 42,5R), Tables 4 and 6, itcan be found that the concentration of the mentionedand other elements (Al and Fe) has increased in con-crete LSLL4 (spectrum 1 and 2-ITZ zone). According toreference [12b] water to cement content have a minorinfluence on the microstructure of the ITZ. In theconditions of the increased water/cement ratio (LSLL 4)

Slika 6. Mesta na kojim je urađ ena EDS analiza za betonLSLL-2 starosti 7 dana

Figure 6. Spots on which EDS analysis was carried outfor 7-day old LSLL-2

Slika 7. Mesta na kojim je urađ ena EDS analiza za betonLSLL-4 starosti 7 dana

Figure 7. Spots on which EDS analysis was carried outfor 7-day old concrete LSLL-4




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što može da se odrazi na povećanje mikrotvrdoće [9]. a continuous dissolution and an interaction of water withCSH happens. The Si content is especially high whichcould increase micro hardness values [9].


Spec O Mg Al Si K Ca Fe Ca/Si Al+Fe/Ca S/Ca


Spec 1 41.52 1.64 2.73 10.53 0.45 42.07 1.06 4.00 0.09 - mix.Spec 2 43.95 0.75 3.05 16.62 0.72 33.59 1.32 2.02 0.13 - CSH


Spec 3 40.58 0.94 4.95 16.70 0.94 33.78 2.12 2.02 0.209 - ≈CSHSpec 4 40.53 0.64 4.13 28.42 1.89 22.48 1.91 0.79 0.267 - mix.

Kod betona gde se kao vezivo koristio cement CEMII/B-M(S-V-L)32,5R, Tabela 5, odnosi elemenataupućuju da se u tranzitnoj zoni (tačka 2) kao i u zrnuLWA-Termit formiraju CSH strukture (Spektar 4, Tabela5, Slika 6 ). Kalcijum-karbonat koji je kao dodatakprisutan u cementu, može da bude veoma efikasan kaonukleus za CSH strukture.

Kod betona LSLL 5 gde je u svojstvu dodatka dodani metakaolin, dobijeni odnosi su slični rezultatima zaprethodno navedene betone. Mnogo veću pažnju privlačismanjenje koncentracije aluminijuma, iako smetakaolinom treba da se poveća sadržaj elementaaluminijuma. U ovom slučaju bi na odvijanje procesahidratacije mogao veću ulogu da odigra polycarboxylate-based superplastificator Sika VSC 4000BP. Absorbcijasuperplastifikatora na površini C 3 A dovodi do

usporavanja procesa hidratacije i povećavanjakoncentracije elemenata Si i Ca, Tabela 7.

In concrete where cement CEM II/B-M (S-V-L) 32,5Rwas used as a binder, Table 5, the ratio of elementsindicates that in ITZ (spectrum 2), as well as in the LWA-Termit grain, CSH structures are formed (Spectrum 4,Table 5, Figure 6). Calcium carbonate from cement isitself quite effective as a nucleus for C-S-H.

In concrete LSLL5 where metakaolin was used as anaddition, the obtained ratios are similar to the results forthe previously mentioned concretes. A much greaterattention is brought to the decrease in concentration ofaluminium, even though with metakaolin the presence ofaluminium should be increased. In this case thedevelopment of the process of hydration could beinfluenced much more by polycarboxylate-basedsuperplastificator Sika VSC 4000BP. With the absorptionon the surface of C3 A compounds the process of the

hydration is slower and the concentration of Si and Ca isincreased, Table 7.


Spec C O Mg Al Si K Ca Ti Fe Ca/Si (Al+Fe)/Ca S/Ca


Spec 1 35.13 0.91 7.55 18.92 0.98 34.45 2.05 1.82 0.279 ≈CSHSpec 2 28.89 7.89 13.27 1.55 46.01 2.40 3.47 0.224 mix.


Spec 3 0.00 31.46 3.58 7.85 0.61 49.39 3.53 3.59 6.3 0.145 mix.Spec 4 11.67 35.09 0.62 4.14 14.27 0.48 32.17 1.55 2.25 0.177 CSH




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Slika 8. Mesta na kojim je urađ ena EDS analiza za beton LSLL-5 starosti 7 dana

Figure 8. Spots on which EDS analysis was carried out for 7-day old concrete LSLL-5

Rezultati EDS analize ukazuju:• da se udeo mešanih produkata hidratacije

povećava dok prisustvo kalcijum- hidroksida nije veliko.Njegovo odsustvo kod betona gde se kao vezivo poredcementa koriste dodaci s pucolanskom aktivnošću(zgura, elektrofilterski pepeo, metakaolin) jesteočekivano (betoni LSLL2 i LSLL5) [5]. Međutim, njegovoodsustvo u drugim uzorcima nije očekivano. Takođe jeinteresantno da nema etringita, s obzirom na to štoprisustvo sumpora nije registrovano. Objašnjenje za ovupojavu može biti interakcija superplastifikatora srastvornim sulfatnim jonima [15].

The results of EDS analysis show• the presence of mixed products of hydration,

whereas the crystals of calcium hydroxide are notregistered in greater amount. The absence/small amountof calcium hydroxide in the samples of concrete whereone of the components were pozzolanic materials wasexpected (concretes LSLL2 and LSLL5) [5]. However, itsabsence in other samples was unexpected. It is alsointeresting that there are no AFt considering that thepresence of sulphur was not found. The explanation forthis could be found in possible interaction ofsuperplastificatros with soluble sulphate ions [15].


Spec O Mg Al Si K Ca Ti Fe Ca/Si Al+Fe/Ca S/Ca


Spec 1 38.95 1.01 3.24 10.19 2.41 42.28 1.92 4.15 0.12 - mix.Spec 2 37.34 3.55 10.05 2.84 44.68 1.54 4.45 0.11 - mix.Spec 3 35.41 3.54 15.27 2.82 41.07 1.88 2.69 0.13 - ≈CSH


Spec 4 45.44 3.46 9.16 2.57 37.75 1.62 4.12 0.135 - mix.Spec 5 43.21 1.27 5.15 11.63 1.37 35.74 1.62 3.07 0.189 - mix.Spec 6 44.26 4.70 13.59 1.28 33.80 0.77 1.60 2.48 0.186 - mix.

• Visoke vrednosti za odnos atoma Ca/Si u TZ zabetone LSLL 1 i LSLL5 upućuju na slabo prisustvokalcijum-siliko-hidrata.

• Promene atomskih odnosa u zrnu LA-Termit uzonama koje su blizu TZ u odnosu na sastav preugrađivanja ukazuje na to da postoji difuzija jona izmeđuzrna i cementne paste. Osim toga, velika hrapavostpovršine zrna LWA-Termit olakšava proces kristalizacijeprodukata hidratacije. Ova pojava je prisutna kod svihanaliziranih betona. U zrnu LA-Termit vrednosti odnosaCa/Si manje su u odnosu na TZ: 1.67 (LSLL-1) i 1.82

• High Ca/Si ratio in ITZ for LSLL-1 and LSLL-5signifies that the ITZ is relatively poor in calcium silicatehydrates.

• The changes of atomic ratios in the LWA-Termitgrain in zones close to ITZ comparing to the compositionbefore the insertion, indicates that there is a diffusion ofions between the grain and the cement paste. Besides, agreat roughness of the surface of the LWA-Termit grainmakes the process of crystallization of hydrationproducts easier. This occurrence is present in all theanalyzed concretes. In LWA-Termit grain the values of






75

Slika 10. Mikrotvrdoć a nakon 28 dana

Figure 10. Micro hardness values after 28 days

4 ZAKLJUČAK

Istraživanja teksturalnih i mikrostrukturnih promena

tranzitne zone korišćenjem SEM i EDS metoda, kao iodređivanje mikrotvrdoće, mogu da doprinesu boljemrazumevanju veze između vrste formiranih produkatahidratacije i jačine veze između agregata i cementnogkamena. Međutim, uspostavljanje direktne veze izmeđuovih vrednosti i makroosobina betona zahteva i dodatnenačine karakterizacije osobina tranzitne zone. RezultatiEDS analize upućuju na to da:

• postoji razmena jona difuzijom između zrna lakogagregata Termit i cementne paste. Osim toga, velikistepen hrapavosti površine zidova pora pomaže procesuheterogene kristalizacije produkata hidratacije i stvaranjaCSH jedinjenja veće gustine;

• vrednosti odnosa Ca/Si velike su za tranzitnu zonu

za sve tipove betona, ali su posebno velike za betoneLSLL-1 i LSLL-5, što ukazuje na to da je TZ relativnosiromašna kalcijum-siliko-hidratima;

• nije moguće dati pouzdanu vezu između sastavatranzitne zone i vrednosti mikrotvrdoće za različitebetone.

ZAHVALNOST

U radu je prikazan deo istraživanja koje je pomogloMinistarstvo prosvete, nauke i tehnološkog razvojaRepublike Srbije u okviru tehnološkog projekta TR36017 pod nazivom: „Istraživanje mogućnosti primeneotpadnih i recikliranih materijala u betonskimkompozitima, sa ocenom uticaja na životnu sredinu, ucilju promocije održivog građevinarstva u Srbiji”.

4 CONCLUSION

The research of textural and micro structural

changes of interfacial transition zone, with the use ofSEM and EDS methods as well as determination ofmicro hardness values, can contribute betterunderstanding of relation between the type of hydrationproducts formed and the strength of the connectionbetween the aggregate and the hardened cement paste.However, in order to make a direct connection betweenthese values and the macro characteristics of theconcrete additional ways of characterization of ITZqualities are needed. The results of EDS analysisindicate that:

• There is an exchange of ions through diffusionbetween the grain of lightweight aggregate Termit andcement paste. Beside that a high level of the roughness

of pore walls aids to the process of heterogeneouscrystallization of the products of hydration and formationof CSH compounds with a greater density.

• Values of Ca/Si ratio are high interfacial transitionzone for all types of concrete, but especially high forLSLL-1 and LSLL-5, signifying that the interfacialtransition zone is relatively poor in calcium silicatehydrates

• It is impossible to make a reliable connectionbetween the composition of the interfacial transition zoneand the micro hardness values for different concretes.

ACKNOWLEDGEMENTS

The research work reported in this paper is a part ofthe investigation within the research project TR 36017"Utilization of by-products and recycled waste materialsin concrete composites in the scope of sustainableconstruction development in Serbia: investigation andenvironmental assessment of possible applications",supported by the Ministry of Education, Science andTechnological Development of the Republic of Serbia.This support is gratefully acknowledged.




76

5 LITERATURAREFERENCES

[1] Asbridge, A.H., Page, C.L., Page, M.M: Effects of metakaolin, water/binder ratio and interfacialtrasition zones on the microhardness of cementmortars, Cement and Concrete Research 32, 2002,1365-1369.

[2] Bonakdar, A., Mobasher, B., Chawla, N.:Diffusivity and micro-hardness of blended cementmaterials exposed to external sulfate attack,Cement & Concrete Composites 34, 2012, 76-85.

[3] Chandra, S., Berntsson, L., Lightweight aggregateconcrete, science, technology and applications, pp11-19, William Andrew Publishing, Norwich, 2002.

[4] Ducman, V., Mirtič, B.: Lightweight aggregateprocesses from waste material, Advances inMaterials Science Research, 4, Chapter 10, 2011,307-323.

[5] Gameiro L.A., Silva S.A., Veiga R. M., VelosaL.A.: Lime-metakaolin hydration products: amicroscopy analysis, Materials and technology 46,

2012, 145-148.[6] Glinicki, A. M., Zielinski, M.: Depth-sensingindentation method for evaluation of efficiency ofsecondary cementitious materials, Cement andConcrete Research 34, 2004, 721-724.

[7] Internal Curing of Concrete, RILEM Report 41,Eds. K. Kovler and O.M. Jensen, RILEMPublications S.A.R.L., 2007.

[8] Li, Y., Wu, D., Yhang, J., Chang, L., Wu, D.,Fang,Y., Shi, Y.,: Measurment and statistics of single pellet mechanical strength of differentlyshaped catalysts, Powder Technology, 113, 2000,176-184.

[9] Lo, Y., Gao, F.X., Jeary, P. A.: Microstructure of pre-wetted aggregate on lightweight concrete,

Building and Environment 34, 1999, 759-764.[10] Mehta, P. K., Monteiro, J.M.P.: Concrete-

Microstructure, Properties and Materials, 3rdEdition, 2006, pp 41-44, Mc-Graw Hill, New York.

[11] Radonjanin, V., Malešev, M., Lukić, I., Milovanović,V.: Polimer-betonski kompoziti na bazi recikliranogagregata, Materijali i konstrukcije, 52, 2009, 1, 91-107.

[12a] Scrivener,L. K., Crumbie, K.A., Laugesen, P..The

interfacial transition zone (ITZ) between thecement paste and aggregate in concrete. InterfaceSci. 12, 2004, 411-421.

[12b] Scrivener, L. K., Characterization of the ITZ andits quantification by test method, in: M. G. Alexander, G. Arlliguie, G. Ballivy, A. Bentur , J.Merchand (eds.), Engineering and TransportProperties of the Interfacial Transition Zone inCementitious Composites, Rilem Report, vol.20,RILEM Publications, 1999, pp 3-14.

[13] Chandra, S.: Properties of concrete with mineralanf chemical admixtures, in: Structure andPerformance of Cements, Edited by J. Benstedand P. Barnes, 2nd Edition, 2001, pp 140-186,Spon Press, London.

[14] Trägárdh, J.:Microstructural features and relatedproperties of self compacting concrete, in: A.Skarendahl, O. Peterson (Eds.). Proceedings ofthe First International RILEM Symposium on SelfCompacting Concrete.RILEM, Cachan, Cedex,1999, .175-186.

[15] Winnefeld, F., Zingg, A., Holzer, L., Pakusch, J.,Becker, S.: Interaction of Polycarboxylate-basedSuperplasticizers and Cements: Influence ofPolimer Structure and C3 A-content of Cement,Cement & Concrete Composites 31, 2009,153-162.

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paste into lightweight aggregate Cement andConcrete Research, 22, 1992, 47-55.




77

REZIME

KARAKTERISTIKE TRANZITNE ZONE BETONA NABAZI LAKOG AGREGATA

Vesna BULATOVICVilma DUCMANMiroslava RADEKA

U radu je prikazana karakterizacija tranzitne zoneizmeđu zrna lakog agregata proizvedenih na baziotpadnih materijala i vezivnog kamena pomoću metodaSEM, EDS i određivanjem Vikersove tvrdoće. Na baziSEM i EDS analize, kao i kriterijuma za odnose atomaelemenata koji ulaze u sastav nekih produkatahidratacije, može se dobiti uvid u preovlađujući sastavtranzitne zone. Vrednosti Vikersove tvrdoće za četirivrste ispitanih betona se nakon sedam dana hidratacijenije moguće povezati sa gustinom pakovanja česticaveziva u tranzitnoj zoni sa velikom sigurnošću, a nakondvadeset osam dana razlike u vrednostima nisuznačajne.

Ključne reči: laki agregat, otpadni materijal,tranzitna zona

SUMM АRY

CHARACTERISTICS OF LIGHTWEIGHT AGGREGATECONCRETE INTERFACIAL TRANSITION ZONE

Vesna BULATOVICVilma DUCMANMiroslava RADEKA

The paper presents the characteristics of theinterfacial zone between the lightweight aggregateproduced on the basis of waste materials and bindermatrix with the application of the methods SEM, EDSand the Vickers micro hardness test. On the basis of theSEM and EDS analysis, as well as the criteria for theatomic ratio of elements which compose some productsof hydration, we can gain insight into the dominantcomposition of the interfacial zone. The values of theVickers micro hardness test for four kinds of testedconcrete after seven days of hydration is impossible tocorrelate with the composition of the interfacial zone inreliable way, whereas after twenty eight days the

differences in values are insignificant.Key words: lightweight aggregate, waste material,

interfacial zone.





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