NUMERICAL SIMULATION OF THE BEHAVIOUR OF THE LIGHTWEIGHT CONCRETE SPECIMEN IN THE LABORATORY TESTING

Beton je veštački kompozitni građevinski materijal dobijen očvršćavanjem mešavine veziva, vode, agregata (i u mnogim slučajevima različitih dodataka) i već duže od jednog veka predstavlja najčešće korišćen građevinski materijal. Budući da godišnja proizvodnja betona premašuje količinu od deset milijardi tona, sve je značajnija upotreba alternativnih materijala. Jedna od mogućnosti upotrebe alternativnih komponenata jeste upotreba lakog agregata, umesto dela ili celokupne količine običnog agregata, kao punioca u betonskoj smeši. Iako je hetegoren, za svakodnevne potrebe projektovanja građevinskih konstrukcija, beton se smatra homogenim materijalom. Tretiranje betona kao homogenog kontinuuma opravdano je kada se beton nalazi u elastičnoj fazi rada, što u najvećem broju situacija i jeste slučaj budući da su konstrukcije u izuzetno retkim slučajevima opterećene graničnim opterećenjima [8]. Napregnute graničnim opterećenjem, armiranobetonske konstrukcije


INTRODUCTION
Concrete is an artificial composite building material obtained by curing the mixture of the binders, water, aggregates (and in many cases various additives) and has been the most commonly used building material for over a century. Since the annual production of the concrete exceeds ten billion tonnes, the use of the alternative materials is of increasing importance. One possibility of using alternative components would be to use a lightweight aggregate instead of a part or the whole amount of a conventional aggregate as a filler in a concrete mixture. Although considered as heterogeneous, for everyday design purposes, concrete is considered to be a homogeneous material. The treatment of concrete as a homogeneous continuum is justified when the concrete is in an elastic phase of operation, which in the most situations is the case, since structures are loaded by the ultimate lads in extremely rare cases [8]. Loaded by the ultimate load reinforce mogu da dožive lom na nekoliko različitih načina: pucanje (prskanje), drobljenje i otpadanje betona, tečenje i izvlačenje zategnutih šipki armature, te tečenje i izvijanje pritisnutih šipki armature. Radi što boljeg opisivanja i sagledavanja ponašanja armiranobetonskih konstrukcija pri lomu, neophodne su napredne numeričke simulacije zasnovane na mehanici kontinuuma. Ovom zahtevu u suprotnosti stoje računarske mogućnosti i vreme koje na raspolaganju imaju inženjeri, te je upotreba pojednostavljenih modela, kako konstrukcije, tako i ponašanja materijala, uobičajena u praksi.
concrete structures can suffer failure in several different ways: cracking, crushing and scraping the concrete, yielding and pulling the tensioned reinforcement bars, and yielding and buckling the compressed reinforcement bars. Advanced numerical simulations based on continuum mechanics are required to better describe and understand the behaviour of the reinforced concrete structures during the failure. This requirement fails to agree with the computational capabilities and the time available to the engineers. Therefore, the use of the simplified models, both of the structure and of the material behaviour, is common in practice.
A great number of research activities has been done in the past few years, both with regard to the development of new, more comprehensive numerical concrete material models and the experimental testing of lightweight concrete. The numerical modelling of the concrete plasticity and the damage development can be performed using the CDP (concrete damage plasticity) material model. This material model introduces the assumption that the two main mechanisms of the concrete failure are cracking due to the tension and crushing due to the compression. This material model has been presented in [4]. It has undergone numerous improvements over the past decade while considering determination of the variables necessary to define it [2,5,9]. Some of the more significant experimental tests on lightweight concrete are given in [7,8,10].
In this paper, the concept of CDP material model is presented and numerical simulation of the behaviour of the lightweight concrete samples in the laboratory testing cases is done. Two test configurations were considered: the determination of the concrete compressive strength on a 150mm concrete cube and the determination of the concrete static modulus of elasticity on a concrete cylinder 150 mm in diameter and 300 mm in height. In order to capture the realistic behaviour of the material, the CDP model of the concrete was used in this paper.

CONCRETE BEHAVIOUR MODEL
The behaviour of the material at failure due to compressive and tensile stress can be described in various ways. In the case of tension, cracks occur, which is reflected in the stress-strain diagram in a sharp drop in the stiffness of the material, followed by a decrease in the stiffness of the load-bearing branch of the diagram. In the case of compression stress, an increase in material volume, crushing and slipping within the material occurs. In combined stress states, failure usually depends on the relation between the principal stresses [2].
The nonlinear behaviour of concrete is described in this paper using a CDP model, implemented in Abaqus © software. This model of non-linear behaviour of materials is primarily intended to model problems where the plasticity of concrete is possible, however, it is also possible to model other quasi-rigid materials. The main assumption is that the two main mechanisms of the concrete failure are cracking due to the tension and crushing due to the compression [1]. The assumption of the material behaviour due to the uniaxial tension and the tensile stress is given in Figure 1. Figure 1. Concrete response to the uniaxial tension (left) and compression (right) [1] Usled jednoosnog zatezanja, veza između napona i dilatacije je linearno-elastična sve dok se ne postigne vrednost čvrstoće betona pri zatezanju σt0. Dostizanje ove vrednosti napona tretira se kao početak formiranja mikroprslina u betonu, koje se prikazuje padom napona, odnosno povećanjem dilatacija.
Due to the uniaxial tension, the stress-strain relation is linear-elastic until the tensile strength of σt0 is reached. Reaching this stress value is treated as the beginning of the formation of micro cracks in concrete, which is represented by a decrease in stress, or by an increase in strains.
Due to uniaxial compression, the stress-strain relationis linear-elastic until the value of the yield stress σc0 is reached. During the plastification, the response is characterized by the stress strengthening, which is then followed by softening the strain after reaching the concrete compressive strength σcu [1].

EXPERIMENTAL TESTING
In this paper, the experimental data of the laboratory tests presented in [7], [8] and [10] were used. The subject of this research was structural lightweight concrete, with the emphasis on the analysis of the most important physical and mechanical properties of fresh and hardened concrete. Five different concrete mixtures were presented in that study, while the results obtained from a single mixture, labelled as the LLK-1, were used in this paper.
Specimens for which the experimental and numerical results were compared in this paper were made from lightweight aggregate, which is based on the expanded clay "Leca-Laterlite" (Italy).The results of the concrete compressive strength tests as well as the determination of the modulus of elasticity of concrete were compared [7].

Configuration 1: Determination of the concrete compressive strength
The compressive strength of the concrete was determined according to the standard SRPS ISO 4012 on cube shaped specimens with the dimension of 150mm, 28 days old, as the mean value of the strengths obtained on three different specimens. The loading rate was 0.6±0.2 MPa/s. The compressive strength test is shown in Fig. 2. The compressive strength of the 28days-old lightweight concrete, determined experimentally, is fcu,28 = 50.6 MPa [8].

Configuration 2: Determination of the concrete modulus of elasticity
The determination of the static modulus of elasticity was done according to the standard SRPS ISO 6784, on specimens of cylinder shape with the dimensions of 150mm in diameter and 300mm in height, between 28 and 35 days old. The test was conducted on three different cylinders, and the representative value of the modulus of elasticity was determined as the mean value of the modulus of elasticity obtained by measuring the stresses and strains on each specimen. The loading rate was 0.6±0.2 MPa/s. Deformations were recorded using an extensometer of the accuracy of 0.001mm with a measuring base of 200mm. The upper load limit (σa≈1/3·fc,28) was defined based on the pre-determined concrete compressive strength, while the lower load limit is defined in the way that the deformation of 0.01mm must be achieved. Specimen testing was performed in two cycles (load series) [7].
where Δllong,a -Δllong,b is expressed in mm as the difference in elongation due to the stresses σa and σb, and 200 mm is the measuring base of the deformeter.
Experimental testing of the modulus of the elasticity is shown in Figure 3. The value of the modulus of elasticity of lightweight concrete determined experimentally is E = 23.21 GPa [7]. Slika 3. Određivanje statičkog modula elastičnosti [10] Figure 3. Determination of the static modulus of elasticity [10]
U oba slučaja je model oslonjen površinski preko donje površine donje čelične ploče, na način da su sprečena pomeranja u sva tri pravca. Trodimenzionalni MKE modeli betonske kocke i cilindra prikazani su na slici 4. cylinder with dimensions d/L = 150/300mm and, as in the previous case, two steel plates with dimensions d/L=160/40mm.Steel plates, located on the top and bottom of the concrete specimens, were used to achieve the real situation, where the cube in the hydraulic press is supported on the bottom on the steel plate, while the load is applied on the upper side, also via steel plate. This allows more realistic modelling of the contact conditions between the specimens and the pedestal. The steel plates thickness of 40mm provides sufficient rigidity for their deformation that should be several times lesser than the deformation of the concrete specimen. In both cases, the model rests over the lower surface of the lower steel plate, in the way that displacements in all three directions are prevented. Three-dimensional FEM models of concrete cube and cylinder are shown in CDP model materijala je korišćen za potrebe modeli-

Material model
Two types of material models were used in the model: the linear-elastic model and the elasto-plastic model (CDP model). A linear-elastic material model was used for modelling steel plates in both configurations. This assumption of material behaviour is justified for several reasons. Firstly, the value of the modulus of elasticity of steel (210 GPa) is more than nine times higher than the value of modulus of elasticity of lightweight aggregate concrete (23.21 GPa). Secondly, the stress values reached in the model are several times lesser than the steel proportionality limit. Therefore, the steel will not enter the plastic phase of the operation at any time. ranja betonskog cilindra i kocke. Ovaj model uvodi pretpostavku o nelinearnom ponašanju materijala. Veza napon-dilatacija lakoagregatnog betona pri pritisku je definisana prema [6] izrazom: The CDP material model was used for the purpose of modelling the concrete cylinder and cube. This model introduces the assumption of nonlinear material behaviour. The stress-strain relationship of lightweight concrete due to the compression is defined according to [6]: (3) gde su E0 inicijalni modul elastičnosti, Es sekantni modul elastičnosti koji odgovara maksimalnom naponu i odgovarajućoj dilataciji, εcu dilatacija koja odgovara maksimalnom naponu, a α parametar koji kontroliše oblik krive, čija je vrednost usvojena prema [6] i iznosi 1.5.
Veza napon-dilatacija betona pri zatezanju usvojena je kao linearna do dostizanja vrednosti čvrstoće betona na zatezanje, nakon čega nastupa lom. Čvrstoća lakoagregatnog betona na zatezanje definisana je prema [3] izrazom: where E0is the initial modulus of elasticity, Es is the secant modulus of elasticity corresponding to the maximum stress and the corresponding strain, εcu is the strain corresponding to the maximum stress, and α is the parameter controlling the shape of the curve, whose value is adopted according to [6] and equals to 1.5.
Here, it is assumed that the stress-strain relationship becomes non-linear after reaching a strain of 5.5·10 -4 , which corresponds to the stress of 12.65 MPa. The tension stress-strain relationship of the concrete is adopted as linear until the tensile strength of the concrete is reached, followed by failure. Tensile strength of lightweight concrete is defined according to [3]:

Finite elements
1D, 2D, 3D and other special, finite elements are available for modelling in Abaqus © . Three-dimensional (solid) FEs were used in this paper. Depending on the type and order of the 3D element, the software offers choices from a 4-node tetrahedral element to a 27-node hexahedral element. Nodes of the three-dimensional elements have 3 degrees of freedom, which refer to three possible translations in space.
In this paper, C3D8R finite elements (3D first order hexahedral KE with 8 nodes) were used. This type of element uses reduced integration to form a stiffness matrix, with the number of integration points being 1 [1].

Contact modelling
The premise that due to the uniaxial compression the specimen fails to suffer any restrictions since transverse deformation does not apply in the area of support of the specimen, but only in the zone that can be approximated by the middle third of the height of the specimen. In the support zones, due to the friction occurring at the taktima uzorak-ploča, dolazi do određenog ograničavanja bočnih deformacija uzoraka. Stoga, naponsko stanje će u tim zonama biti znatno složenije u odnosu na uslove aksijalnog pritiska.
Vremenske funkcije opterećenja za obe konfiguracije ispitivanja date su na slici 5. Nanošenjem opterećenja u intervalima od 100 s i 27 s aproksimirano je realno stanje u kojem se opterećenje koje izaziva napone u presecima kocke i cilindra u iznosu od 50.3 MPa i 13.5 MPa nanosi brzinom od 0.6±0.2 MPa/s. specimen-plate contacts, there is some limitation of the lateral deformations of the specimens. Therefore, the stress state in these zones will be much more complex than the axial compression conditions. The contact between specimens (cubes; cylinders) and steel plates was modelled using tangential and normal components to meet this argument. Coefficient of friction of 0.57 was assigned to the tangential component (friction at the specimen-steel plate surface), which is the common value of the friction coefficient between the concrete and steel. The normal component is modelled in the way that, when the surfaces are in contact, any contact compression between them can be transferred, whereby the contact surfaces are separated if the contact compression reaches zero. In other words, tension on the contact surface in normal direction is not allowed.

Load definition
The load is defined in both cases (cube test and cylinder test) as uniformly distributed surface load and acts on the upper surface of the upper steel plate.
The time load functions for both test configurations are given in Figure 5. Applying a load at 100 and 27 s intervals approximates the real state in which the load that causes stresses in the cube and cylinder sections of 50.3 MPa and 13.5 MPa, respectively, is applied at the rate of 0.6 ± 0.2 MPa/s. Figure 5. Time function of the load of concrete cube (left) and concrete cylinder (right)

ANALYSIS OF THE RESULTS OF THE NUMERICAL MODEL
Nonlinear static analyses were performed for the two configurations of numerical models. Newton's method was used to iteratively solve the system of equations, with the maximum number of increments being 100.
The results of the nonlinear static analysis are given only for the part of the numerical model that includes concrete specimens, since the steel plates in the numerical model are present for the most accurate representation of the contact conditions in the contact zone with the specimen. Due to their thickness and the fact that their modulus of elasticity is several times larger, as well as the yield strength, steel plates remain in the elastic phase during the analysis, and will not be considered in this paper.

Configuration 1
The results of the cube model show the displacements, stresses and corresponding strains of the specimen, as well as the shear stresses at the contact between the specimen and the steel plate. Figure 6 shows the deformed shape of the specimen in the side view with the values of the vertical displacement, while Figure 7 shows the lateral displacement of the specimen in the X direction.
It is noticeable that the specimen deforms in the way that corresponds to the real state: since, due to uniaxial compression, the specimen does not suffer any restrictions in terms of transverse deformation. It freely deforms in the transverse direction in the zone of the middle third of the specimen. When approaching the supports, the transverse deformation is reduced due to the friction that occurs on the specimen-plate contacts. The maximum vertical displacement of the specimen is: U2 = 0.8124 mm -0.0137 mm = 0.7987 mm. Figure 8 shows the stresses and corresponding strains in the longitudinal direction of the cube (in height).
Slika dilatacija prati sliku napona. Maksimalne dilatacije na spoljašnjoj strani imaju vrednosti 3.150·10 -3 i -5.547·10 -3 a u unutrašnjosti 3.494·10 -3 i -7.160·10 -3 . directions in the cross-section (vertical section) of the cube. Here, a clear distinction is made between the tensiled and the compressed sections of the concrete cube, which occupies a spatial X-shape. This type of failure occurs due to the friction forces on contact with the steel plates, which confine the concrete. The confined part of the concrete is a wedge-shaped part in the lower and upper parts of the cube, which acts in such a way to squeeze out the surrounding tensiled concrete after reaching the tensile strength of the concrete and formation of the first cracks.
Higher values of stress inside the cube are due to the confinement of the concrete, i.e. the fact that the concrete closer to the outside prevents lateral deformation of the concrete inside the cube.

Configuration 2
The results of the cylinder model show the displacements of the specimen in the vertical and lateral directions, the stresses in the longitudinal direction of the cylinder axis and the corresponding strains, the strains in the transverse directions, and the shear stresses at the contact between the specimen and the steel plate. Figure 12 shows the deformed shape of the specimen in the side view with the values of the vertical displacement, while Figure 13 shows the lateral displacement of the specimen in the X direction Može se zapaziti sličnost u pogledu oblika deformacije uzoraka cilindra i kocke: slobodna poprečna defor- The similarity may be observed with respect to the deformation patterns of the cylinder and cube speci-macija cilindra usled jednoosnog pritiska u zoni srednje trećine uzorka. Približavanjem osloncima poprečna deformacija smanjuje se usled trenja koje se javlja na kontaktima uzorak-ploča. Maksimalno vertikalno pomeranje uzorka iznosi: U2 = 0.2482 mm -0,0027 mm = 0.2455 mm.
Na slici 14 dati su slika i vrednosti napona u podužnom pravcu cilindra, na jednoj polovini uzorka i na celom uzorku. Slike 15 i 16 daju prikaz slike i vrednosti podužnih i poprečnih dilatacija uzorka. mens: free transverse deformation of the cylinder due to uniaxial compression in the mid-third zone of the specimen. When approaching the supports, the transverse deformation is reduced due to the friction that occurs on the specimen-plate contacts. The maximum vertical displacement of the specimen is: U2 = 0.2482 mm -0.0027 mm = 0.2455 mm. Figure 14 shows the pattern and the values of the stress in the longitudinal direction of the cylinder, on the one half of the specimen and on the entire specimen. Figures 15 and 16  Na slici 17 prikazani su naponi u ravni kontakta cilindra i čelične ploče s gornje strane.
The maximal stress on the outside of the cylinder is 15.09 MPa and 15.91 MPa on the inside of the cylinder. The compression values inside the cylinder are higher than the outer parts for the same reason described in the previous configuration.
Here, as in the previous configuration, the strain pattern follows one of the stress. The maximum strain on the outside of the cylinder is 8.336·10 -4 and 8.733·10 -4 on the inside. Figure 17 shows the stresses in the contact surface between the cylinder and the steel plate on the upper side. Figure 17. Shear stress [MPa] at the contact between the concrete cylinder and the steel plate (half of the specimen): a) in the X direction and b) in the Y direction

Rekapitulacija rezultata
U Tabeli 2 prikazani su najznačajniji rezultati laboratorijskih ispitivanja i numeričkih analiza. Table 2 shows the most significant results of the experimental testing and the numerical analyses.

CONCLUSIONS
The experimental data from the laboratory tests presented in [7], [8] and [10] served as the basis for the formation of the FEM model of the behaviour of the lightweight concrete. The experiment determined the compressive strength of the concrete and the modulus of elasticity of concrete at 28 days of age for several different mixtures, while numerical analysis included one mixture (LLK-1). The behaviour of concrete as a material was accounted by the concrete damage plasticity (CDP) model. The static nonlinear (incremental-iterative) analysis was conducted for two model configurations, one for the determination of the compressive strength of concrete and the other for the determination of the modulus of elasticity of concrete. The first configuration of the model consists of the cube measuring 150x150x150 mm and the second one of the cylinder d/L = 150/300mm. The results of the numerical analyses were analyzed and the following conclusions were drawn: • Concrete cube behaviour in FEM analysis is close to cube behaviour during laboratory testing. Namely, as in the experiment, the failure mechanism pattern is clearly expressed in the numerical model as well. This shape takes the spatial X-shape, since due to the friction forces on the contact with the steel plates, a wedgeshaped portion of concrete is formed, which acts to squeeze out the surrounding tensiled concrete after reaching the tensile strength and forming the first cracks. The compressed concrete has aforementioned X-shape, while in the concrete closer to the outside the tension occurs, so that the boundary between the zones of these two stress states is clearly visible. It is at this border that cracks appear that will impair the integrity of the specimen and dictate the shape of the failure.
• The results of the numerical model in the overall view indicate a congruence with the experimental results. However, further fine-tuning of the FEM model, especially the concrete behaviour model, is required in order to be fully compatible with the experimental data.
• The behaviour of the concrete cylinder in the FEM analysis is, as in the case of the concrete cube, close to the behaviour of the cylinder during laboratory testing. The 0.245 mm cylinder shortening in the FEM model is very close to the average cylinder shortening in the experiment, which is amounted to 0.218 mm, at the same load level.
• The deformation of the cylinder in the FEM model approximates the real state, where due to uniaxial compression in the middle third zone of the specimen, the specimen fails to suffer any restrictions in terms of • Takođe, dilatacija na spoljašnjoj strani cilindra, u srednjoj trećini visine, u MKE modelu ima vrednost od 0.85·10 -3 , dok je u toj istoj zoni u eksperimentu registrovana dilatacija od 1.10·10 -3 .
• Veće vrednosti napona u unutrašnjosti cilindra (15.91 MPa) u odnosu na vrednosti napona na transverse deformation, and it deforms freely in the transverse direction. In the contact zone with the steel plates, the transverse deformation is reduced due to the friction that occurs on the specimen-plate contact surface.
in the middle third of the height, has a value of 0.85·10 -3 in the FEM model, while a strain of 1.10·10 -3 was registered in the same zone in the experiment. Higher values of stress inside the cylinder (15.91 MPa) compared to the stress values on the outside of the cylinder (15.09 MPa) in the FEM model are due to the confinement of the concrete, i.e. the fact that the concrete closer to the outside prevents lateral deformation of the concrete inside the cylinder. In addition, the reasons for higher stress values in the FEM model compared to the experiment can be found in the fact that in the experiment the stress was determined as a quotient of the total force and the cross-sectional area of the cylinder, while in the FEM analysis the cylinder was considered as a three-dimensional body, with the possibility of consideration of the triaxial compression state.