The subject of this paper is to determine the idealised stress-strain diagram (primal curve) as well as a numerical expression of a straight line and a parabolic curve for glass fibre reinforced polymers (GFRP). The intention is to create a tool, by which one could define the stress-strain diagram for any given glass fibre based on tensile strength. GFRP is a material that is increasingly being used in structures. Because of that we need to know some of its basic characteristics or of other similar composite materials, so as to be able to calculate these structures. This paper presents the modulus of elasticity determined by testing glass fibre reinforced polymer samples and a proposition of an idealised stress-strain diagram as well as a numerical determination of the modulus.Predmet ovog rada je određivanje idealiziranog dijagrama naprezanje-deformacija (primarna krivulja) te brojčana definicija pravca i krivulje drugog reda za stakloplastiku (GFRP). Namjera je, također, da ovaj rad postane alat kojim bi se mogao definirati dijagram naprezanje-deformacija za bilo koju drugu stakloplastiku na osnovu rastezne čvrstoće. Stakloplastika je materijal koji je ušao u širu primjenu u graditeljstvu, što zahtjeva i poznavanje određenih karakteristika same stakloplastike ili sličnih kompozitnih materijala, kako bismo mogli proračunati te konstrukcije. U ovom radu dan je modul elastičnosti za stakloplastiku određen ispitivanjima te prijedlog idealiziranog radnog dijagrama stakloplastike i računskog određivanja modula elastičnosti

Aleksandar Jurić

Pavao Marović

Tihomir Štefić

The subject of this paper is to determine the idealised stress-strain diagram (primal curve) as well as a numerical expression of a straight line and a parabolic
 curve for glass fibre reinforced polymers (GFRP). The intention is to create a tool, by which one could define the stress-strain diagram for any given glass fibre
 based on tensile strength. GFRP is a material that is increasingly being used in structures. Because of that we need to know some of its basic characteristics or of
 other similar composite materials, so as to be able to calculate these structures. This paper presents the modulus of elasticity determined by testing glass fibre
 reinforced polymer samples and a proposition of an idealised stress-strain diagram as well as a numerical determination of the modulus.Predmet ovog rada je određivanje idealiziranog dijagrama naprezanje-deformacija (primarna krivulja) te brojčana definicija pravca i krivulje drugog reda za stakloplastiku (GFRP). Namjera je, također, da ovaj rad postane alat kojim bi se mogao definirati dijagram naprezanje-deformacija za bilo koju drugu stakloplastiku na osnovu rastezne čvrstoće. Stakloplastika je materijal koji je ušao u širu primjenu u graditeljstvu, što zahtjeva i poznavanje određenih karakteristika same stakloplastike ili sličnih kompozitnih materijala, kako bismo mogli proračunati te konstrukcije. U ovom radu dan je modul elastičnosti za stakloplastiku određen ispitivanjima te prijedlog idealiziranog radnog dijagrama stakloplastike i računskog određivanja modula elastičnosti

Štefić, Tihomir

Jurić, Aleksandar

Marović, Pavao

Repository of Faculty of Civil Engineering Osijek

1IntroductionUvodThe progress in civil engineering has caused a wider useof modern composite materials, as it has already been thecase with reinforced concrete. Composites used today arenormally made of a matrix and some kind of filler thatreinforces the material. The filler or the reinforcement isusually a light, strong and brittle material and the filler ismostly a material with more elastic characteristics, whosefunction is to bond, to transfer the loads and to protect thefiller. The application and analysis of composite materialswas the subject of the papers by many authors [1, 4, 5], aswell as by authors of books and manuals [2, 3].The basic characteristic of a material, seen through theHooke's law, is the modulus of elasticity, known as theYoung's modulus. To put it shortly, it represents the ratio ofstress and strain ( = / ). This value is defined in the elasticarea. For materials that have a distinct plastic area it is usualfor the value to be known only in the elastic area. By usingan idealised stress-strain diagram, which is normallydefined by straight lines, the elastic area where values of themodulus are exact and the area of plastic behaviour wherethe value of the modulus gradually decreases, is described.All materials have both elastic and plastic area. In order tosimplify any calculations, one value of the modulus is used,and for the plastic area it is normally defined to begin whenthe irreversible deformations reach the value of, forexample 0,1 %.The modulus of elasticity is considered to be the basiccharacteristic of a material and it is used for mostcalculations of any structures, so it is important to know itsvalue. Composite materials have a long history ofapplication in civil engineering and in recent history wehave been using composites like carbon, teflon, glass fibrereinforced polymers, concrete reinforced with carbonE σ ε69T. Štefić, A. Jurić, P. MarovićISSN 1330-3651UDC/UDK 539.37:678.017DETERMINATION OF MODULUS OF ELASTICITY FORGLASS FIBRE REINFORCED POLYMERSTihomir Štefić, Aleksandar Jurić, Pavao MarovićThe subject of this paper is to determine the idealised stress-strain diagram (primal curve) as well as a numerical expression of a straight line and a paraboliccurve for glass fibre reinforced polymers (GFRP). The intention is to create a tool, by which one could define the stress-strain diagram for any given glass fibrebased on tensile strength. GFRP is a material that is increasingly being used in structures. Because of that we need to know some of its basic characteristics or ofother similar composite materials, so as to be able to calculate these structures. This paper presents the modulus of elasticity determined by testing glass fibrereinforced polymer samples and a proposition of an idealised stress-strain diagram as well as a numerical determination of the modulus.Keywords: glass fibre reinforced polymer (GFRP), modulus of elasticity, primal curve, stress-strain diagramcomposite, elastoplastic material,Preliminary notesPredmet ovog rada je idealiziranog dijagrama nap -deformacija (primarna krivulja)(GFRP). Namjera je, t , da ovaj rad postane alat kojim bi se mogao definirati dijagram nap -, što zahtjeva i poznavanjesmote prijedlog idealiziranog radnog dijagrama stakloodređivanje rezanje te brojčana definicija pravca i krivulje drugog reda zastakloplastiku akođer rezanje deformacija za bilo koju drugustakloplastiku na osnovu rastezne čvrstoće. Stakloplastika je materijal koji je ušao u širu primjenu u graditeljstvu određenihkarakteristika same stakloplastike ili sličnih kompozitnih materijala, kako bi mogli proračunati te konstrukcije. U ovom radu dan je modul elastičnosti zastakloplastiku određen ispitivanjima plastike i računskog određivanja modula elastičnosti.Ključne riječi: dijagram nap deforma kompozit, primarna krivulja, stakloplastika (GFRP)rezanje- cija, elastoplastični materijal, modul elastičnosti,Prethodno priopćenjeOdređivanje modula elastičnosti stakloplastikeOdređivanje modula elastičnosti stakloplastikeTechnical Gazette 18, (2011),1 69-72fibres, rebar and similar. Because of the constant use of newmaterials there is a need to define some basic characteristicsof a material through, for example, tensile strength.The data from the testing of samples is taken from theexperiments of glass fibre reinforced polymers from theliterature [1], for two similar samples broken with tensileforce. The samples showed elastic behaviour up to one thirdof the breaking value, and on the remaining two thirds thematerial behaved plastically. That is, for every increase ofthe force the value of the modulus decreased slightly. Andafter the release of the force and the reapplying, the plasticdeformation started to reoccur with the increase of the forcein comparison with the previous value. In total, two sampleswere observed and they showed similar shapes of the stress-strain diagrams and modulus of elasticity.The shapes of the samples were taken according to theAmerican Standard ASTM D 3090, as shown in Fig. 1. Themachine used for the testing is described in more detail inthe literature [1], and is shown in Fig. 2.2The testing of samplesIspitivanje uzorakaR7665 41,840156541,8116,5330Figure 1Slika 1.Geometry of the tested GFRP sample (units in mm)Geometrija ispitivanog GFRP uzorka (jedinice u mm)70Determination of modulus of elasticity for glass fibre reinforced polymersTehni ki vjesnikč ,18, 1(2011)  69-72T. Štefić, A. Jurić, P. MarovićThe mean value of the modulus of elasticity for theobserved samples is 66,4 N/mm , for the elastic area, and itfalls to around 20 N/mm prior to the breaking point. If weconstruct a diagram using the data for the elastic area, as aline, which would represent the behaviour of these samples,then for the sample 1 it would look as shown in Fig. 7 (forbetter appearance additional 3,5 N/mm were added on thestress axis), and the mathematical expression is also shownin Fig. 7. Furthermore, this was also done for the plastic areaof all the increments, together with the mathematicalexpression, and shown in Fig. 7 (also with additional 3 5N/mm on stress axis). One irregularity is visible in Fig. 5, inthe shape of the line deflection at the strain value of 0,2 %and stress value of about 15 N/mm . This can be attributed tothe slipping of the samples at the attachment points. In Fig.8, which represents the second sample, additional 3,5N/mm was also added on the stress axis for better visibility.222222,3The determination of the modulus of elasticity for glassfibre reinforced polymerOdređivanje modula elastičnosti stakloplastikeThe data from the testing have shown that glass fibrereinforced polymers behave as an elastic material up to acertain value of stress, and after that they start to deformplastically all the way to the breaking point. Thecharacteristic of this material is that after every increase ofstrain, that is, after every increment of the testing procedure,the plastic deformation begins at the value at which theprevious increment finished. Knowing this, it is possible todivide the stress strain diagram into two areas. The elasticarea expands up to one third of the bearing value. In this areathe material does not exhibit irreversible deformations, thatis, it behaves as an elastic material (Fig-s 3 and 5). Afterexceeding this value the material starts to show irreversibledeformations. The line that describes the relaxing of thesamples appears to be fairly linear, which implies theassumption that the material does not need time to take theoriginal shape, at least from the elastic area (Fig-s 4 and 6).It is also visible that the ultimate strength of these samples isroughly 60-65 N/mm , at which time the material exhibitsstrain in the value of about 0,2 %.2-Figure 2Slika 2.Testing machine with the samplesIspitni stroj s uzorcima-100102030405060-0,2 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6Strain, %1.increment 2.increment 3.increment 4.increment5.increment 6.increment 7.increment 8.incrementStress, N/mm2Figure 3Slika 3.σ εσ-ε dijagram, uzorak 1, 8- diagram, sample 1, 8 increments, applying stresspriraštaja, primjenom naprezanja-100102030405060-0,2 0 0,2 0,4 0,6 0,8 1 1,2Strain, %1.increment 2.increment 3.increment 4.increment5.increment 6.increment 7.incrementStress, N/mm2Figure 4Slika 4., sample 1, 7 increments, releasing stress7 priraštaja, naprezanjaσ-ε diagramσ-ε dijagram, uzorak 1, oslobađanjemStress, N/mm2-55152535455565-0,2 0 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6Strain, %1.increment 2.increment 3.increment4.increment 5.increment 6.incrementFigure 5Slika 5.σ-ε diagramσ-ε dijagram, uzorak, sample 2, 6 increments, applying stress2, 8 priraštaja, primjenom naprezanjaIt is visible that the - diagram for fibre reinforcedpolymer consists of a straight line, which represents theelastic area, where the stress and strain have a linearrelation, and that this area reaches up to 1/3 of the breakingstress. After that, 1/3 of the area follows the 2/3 plastic area,which is fairly approximated by a second order curve.σ ε71Technical Gazette ,18, 1(2011)  69-72T. Štefić, A. Jurić, P. Marović Određivanje modula elastičnosti stakloplastikeThe determination of the modulus of elasticity in thearea of 1/3 breaking stress is simple and can be seen fromFig-s 7 and 8. If there is a need to know the value of theelasticity modulus for a specific stress of a material, (forexample the stress at which the material already sufferedsome plastic deformation), this could be done by solving thesecond order equation, that also describes the diagram curvein Fig-s 7 and 8.If we assume that is for a small portion larger thanis 0,1 N/mm , the strain can becalculated by solving the equation (3), so the following canbe written:σ σσ2 1,or in other words that Δ2Stress, N/mm2-15-55152535455565-0,1 0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8Strain, %1.increment 2.increment 3.increment 4.increment 5.incrementFigure 6Slika 3.σ-ε diagramσ-ε dijagram, uzorak, sample 2, 5 increments, releasing stress2, 5 priraštaja, naprezanjaoslobađanjemStress, N/mm2y = -19,5x2 + 67,6x + 6y = 69,3x + 3,8y = -19,465x2 + 67,614x + 2,5235051015202530354045505560650 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6Strain, %Idealised diagram- Plastic area Idelised diagram- Elastic areaSample 1 only plastic area Poly. (Idealised diagram- Plastic area)Linear (Idelised diagram- Elastic area) Poly. (Sample 1 only plastic area)Figure 7Slika 7.Idealised diagram – sample 1Idealizirani dijagram – uzorak 1σ-εσ-εStress, N/mm2y = -18,565x2 + 67,749x + 1,6095y = -18,6x2 + 67,7x + 5,1y = 63,6x + 3,50102030405060700 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6Strain, %Sample 2 only plastic area Idealised diagram - Plastic areaSample 2 only elastic area Poly. (Sample 2 only plastic area)Poly. (Idealised diagram - Plastic area) Linear (Sample 2 only elastic area)Figure 8Slika 8.Idealised diagram – sample 2Idealizirani dijagram – uzorak 2σ-εσ-ε219 5 67 6 2 5y , x , x ,      (1)2 12        1E .    (2)21 242,b b a cxa 	   (3)     221 267 6 67 6 4 19 5 2 52 19 5,, , , ,,  	     (  )4     221 267 6 67 6 4 19 5 0 1 2 52 19 5,, , , , ,.,  	      (  )5Combining equation (2) with equations (4) and (5) weobtain next formula:           2 22 201676 676 4 195 676 676 4 195 012 2,E, , , c , , , , ca a   	        	      (  )6from which we can define the modulus of elasticity for anyspecific stress in the plastic area of glass fibre reinforcedpolymers. As the correct solution for strains , , and, for any strain the positive result is chosen. Shown inTab. 1 is a significantly different modulus of elasticitycalculated for several stresses in the plastic area.ε ε εε1,1 1,2 2,12,2Table 1Tablica 1.Calculation of several modul of elasticity for differentstresses in the plastic area for sample 1esProračun nekoliko modula elastičnosti za različitanaprezanja u području plastičnosti za uzorak 1N/mm 235 40 45 50 55 60a b c0,1 19,5 -67,6 -2,535,5 29,5 21,9 9,4N/mm2EN/mm 245,2 40,6y = ax 2 + bx + c- ax 2- bx - (c + y) = 0Stress, N/mm2y = 31,455x + 16,353051015202530354045505560650 0,5 1 1,5 2Strain, %Plastic area Elastic area Bilinear - plastic area Linear (Plastic area)Figure 9Slika 9.Idealised bilinear diagramσ-εIdealizirani bilinearni σ-ε dijagramIn order to simplify even more the idealised stress-strain diagram, it is possible to present it as a diagramconsisting of two linear lines, i.e. bilinear diagram, asshown in Fig. 9.When looking at the bilinear approximation it must beconsidered that the value of modulus of elasticity is alsoconstant in the plastic area, while it is visible that after 2/3 of72 Tehni ki vjesnikč ,18, 1(2011)  69-72breaking force (or stress) the value of modulus falls underthe approximated value. In other words, this kind of adiagram understates the value of from 1/3 to 2/3, andoverstates the value of above the 2/3 of the breaking force.According to these statements, the use of linear -parabolic idealised diagram shown in Fig. 10 isrecommended.EEDetermination of modulus of elasticity for glass fibre reinforced polymers T. Štefić, A. Jurić, P. MarovićStress, N/mm20510152025303540455055606570750 0,2 0,4 0,6 0,8 1 1,2 1,4 1,6 1,8Strain, %Plastic area Elastic area Sample 2 Plastic area Sample 1 Plastic areaFigure 10Slika 10.Comparisonjagramaof different σ-ε diagramsσ-ε diUsporedba različitihIf the mean value of the two samples is considered, itcan be said that the behaviour of glass fibre reinforcedpolymers in the elastic area or the value of the modulus ofelasticity can be described by the following equation:66,4y x.  (  )7The plastic area is approximated by a curve of thesecond order and if the mean value of two samples isconsidered, it can be said that the behaviour of glass fibrereinforced polymer in the plastic area or the value of themodulus of elasticity can be described by the followingequation:(  )8219 05 67 65 2 05y , x , x , .     4ConclusionZaključakIt has been shown that on the basis of tensile strength ofglass fibre reinforced polymers the value of the modulus ofelasticity can be determined for any area (elastic or plastic)the material is in. The division is made in elastic and plasticarea. Elastic area is approximated by a straight line and it issituated in the first third of the bearing strength of thematerial. The remaining area, until the breaking point of thematerial, is the plastic area.According to the results we recommend the use of thelinear - parabolic shape of the idealised stress-straindiagram, whose shape describes the testing results verywell, as shown in Fig. 10. In addition to that, there is anobligatory division of the stress-strain diagram in elasticarea, which goes till 1/3 of the tensile strength, and plasticarea, which goes from 1/3 of tensile strength to the breakingpoint of the material.5ReferencesLiteratura[1] Štefić, T. Jurić, A. Maričić, S.a i mjeračima pomaka(LVDT-ima)) Treći internacionalni naučno-stručni skupGrađevinarstvo-nauka i praksa (2010), 759-764[2] Šimunić, Ž.[4] Lacković, V. Šimić, V.Ponašanje kompozitnihmaterijala pri složenom opterećenju Građevinar; ; Strength of glass fiberreinforced polymers tested using strain gauges anddisplacement meters (LVDT-S) (Nosivost uzoraka odstakloplastike ispitivana tenzometrim. //(in Croatian).Polymers in Civil Engineering (Polimeri ugraditeljstvu), University of Zagreb, Faculty of CivilEngineering, Zagreb, 2006. (in Croatian).[3] Gerdeen, J. C.; Lord, H. W.; Rorrer, R.A. Engineering Designwith Polymers and Composites. // CRC Press, Taylor &Fancis Group, Boca Raton, 2006.; Behaviour of composite materialssubjected to combined load (). // , 58,7(2006), 549-557 (in Croatian).[5] Hollaway, L.C. The evolution of and the way forward foradvanced polymer composites in the civil infrastructure. //Construction and Building Materials 17 (2003), 365–378.Authors' addressesAdrese autoraTihomir Štefić, dipl. ing. građ.Doc. dr. sc.Aleksandar Jurić, dipl. ing. građ.Prof. dr. sc. Pavao Marović, dipl. ing. građ.J. J. Strossmayer University of OsijekFaculty of Civil Engineering OsijekDrinska 16a31000 Osijek, Croatiae-mail: tstefic@gfos.hrJ. J. Strossmayer University of OsijekFaculty of Civil Engineering OsijekDrinska 16a31000 Osijek, Croatiae-mail: ajuric@gfos.hrUniversity of SplitFaculty of Civil Engineering andArchitectureMatice hrvatske 1521000 Split, Croatiae-mail: marovic@gradst.hr

Određivanje modula elastičnosti stakloplastike

Abstract

Similar works

Full text

Available Versions

Repository of Faculty of Civil Engineering Osijek

Hrčak - Portal of scientific journals of Croatia

Repository of Josip Juraj Strossmayer University of Osijek