A mathematical model for complete stress-strain curve prediction of permeable concrete

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1 A mathematical mdel fr cmplete stress-strain curve predictin f permeable cncrete M. K. Hussin Y. Zhuge F. Bullen W. P. Lkuge Faculty f Engineering and Surveying, University f Suthern Queensland, Twmba, Australia ABSTRACT: An empirical equatin t represent the cmplete stress-strain behaviur fr uncnfined permeable cncrete with cmpressive strength ranging between 10-35MPa and prsity ranging between 25-15%, made with different cmbinatins f aggregate size and sand ratis is prpsed in this paper. A series f cmpressin tests were cnducted n mm cylindrical samples using a mdified testing methd t determine the cmplete stress-strain behaviur f permeable cncrete. Varius existing mdels fr lw strength cncrete and nrmal strength cncrete were used and cmpared with the experimental data. Varius parameters were studied and their relatinships were experimentally determined. The nly parameters need t run the mdel is the ultimate cmpressive strength and the density. The prpsed empirical stress-strain equatins were cmpared with actual cylinder tests results under axial cmpressin, and demnstrated that the present mdel gives a gd representatin f the mean behaviur f the actual stress-strain respnse. 1 INTRODUCTION There has been an increasing use f permeable cncrete in the civil engineering and building cnstructin industries in recent years (Offenberg 2008). Hwever its use is currently limited t lw trafficked areas such as pavements in car parks and ftpaths, largely due t its lw strength and stiffness. It is timely t investigate the stress-strain behaviur f permeable cncrete t help enable its wider use in mre structural applicatins. An understanding f the cmplete stress-strain curve f permeable cncrete is essential fr ratinal design, as structural designers are unable t take full advantage f the material with insufficient infrmatin abut this behaviur. A number f researchers (Attard & Setunge 1996,Carreira & Chu 1985, Kent & Park. 1971, Kumar 2004, Lkuge et al. 2004, Lkuge et al. 2005, Ppvics 1973, Sargin et al. 1971, Tasnimi 2004, etc) have studied the stress-strain behaviurs f uncnfined and cnfined cnventinal cncretes under uni-axial cmpressive lading. Cmplete stress-strain relatinships were develped in sme f these studies n the basis f easily measured cnventinal cncrete material parameters such as peak stress f c, the crrespnding strain ε and the initial mdulus f elasticity E. The infrmatin abut the stress-strain behaviur f the permeable cncrete is still limited (Hussin et al. 2012) and the nly research paper that reprted this behaviur was based n experiments cnducted by De & Neithalath (2010). These authrs fund that the stress-strain behaviur f the permeable cncrete is apprximately similar t that f the cnventinal cncrete. It is reprted that there were many material parameters such as aggregate size, prsity rati, and the pre structural features, as well as the test methd that influenced stress-strain behaviur. De & Neithalath (2010) prpsed a mathematical stress-strain mdel fr the permeable cncrete that was related t the pre structure. De & Neithalath (2010) emplyed the mdel prpsed by Carreira & Chu (1985) fr the cnventinal cncrete with sme mdificatins. The prpsed empirical mdel was fund t agree satisfactrily with their wn test results but it required extensive cmputatins t determine the essential parameters and cannt be used t represent the stress-strain behaviur f a different strength f permeable cncrete that is prepared under different cnditins. Hwever, the ascending branch f the stress-strain curve is well represented by the mdel; and mst discrepancies between the actual and the predicted curve were bserved in the descending branch. 2 PROPOSED STRESS-STRAIN RELASHIONSHIP In an effrt t cnstruct a simple mathematical mdel that can represent the stress-strain curve f permeable cncrete with different cmpressive strengths and prsity ratis, it was fund that the

2 mdel develped by Carreira & Chu (1985) and later mdified by different authrs such as Tasnimi (2004) is suitable fr adpting in the stress-strain relatinship f permeable cncrete. The advantages f this mdel are: It perfrms well fr the lw strength and nrmal strength cncrete, which are bth in the same range as the permeable cncrete s cmpressive strength. It is simple and has the same general equatin fr use in bth the ascending and the descending branch. Its parameters are easy t find frm the experimental data. Equatin 1 illustrates the general equatin fr the stress-strain behaviur as prpsed by Carreira & Chu (1985). ' ( / ) f f c c (1) [ 1( / ) ] test specimens were autmatically cllected by the attached cmputer. Preliminary lw level lading was carried ut at least three times primarily fr the seating f the gauges and fr investigating and crrecting any unusual behaviur f the strain gauge accrding t AS (1997). Figure 1. Permeable cncrete sample with strain gauges. 1 ' f 1 ( c E ) (2) Where: f c = cncrete stress; f c = maximum stress; β = material parameter; ε =cncrete strain; ε =crrespnding strain at maximum stress; and E = initial tangent mdulus f elasticity. 3 EXPERIMENTAL PROGRAM Twenty six specimens with different cmpressive strength and prsity ratis were tested. Strain gauging and the platen-t-platen methds were used t find the stress-strain relatinship. Befre testing, the cylinders were capped with a sulphur cmpund n bth ends t prduce a smth surface t ensure unifrm transfer f lad. Tw diametrically ppsite 60 mm lng strain gauges, were attached t each specimen in its middle third psitin. A prepared specimen ready t be tested is shwn in Figure 1. The signals frm the tw strain gauges were averaged fr a mre accurate result. The results frm the strain gauges were used t find the ascending branch f the stress-strain curve until the peak stress; the strain gauge wuld nt give reliable results after that due t the develpment f vertical cracks in the surface f the specimens. The residual parts f the curve were determined using the platen-t-platen methd. An Avery 500kN testing machine was emplyed fr this purpse and the experimental set-up is shwn in Figure 2. The strain rates f the test specimens were kept cnstant t 10µs per secnd in rder t btain the stress-strain curves. During the experiment, the axial cmpressin lad and the vertical defrmatin f the Figure2. Experimental set-up 4 PREDICTION OF THE MODEL PARAMETERS 4.1 Estimating strain at peak stress (ε ) One f the mst imprtant parameters affecting the ascending and descending prtins f the cncrete stress-strain curve is the crrespnding strain at peak stress. Table 1 prvides the values f the crrespnding strain at peak stress, the initial mdulus f elasticity and the density ρ frm ur testing results. As similar tests were cnducted by Gede (2009), his results were als included. The maximum and minimum values f ε fr the permeable cncrete specimens in this research varied frm t , the crrespnding cmpressive strength varied frm 9MPa t 38MPa and the prsity varied frm 25% t 15% respectively. The tests results indicated that, n average, the peak strain increased as cmpressive strength increased and the prsity rati decreased.

3 Table 1 Experimental results Sample f c ε ρ E it N MPa % kg/m 3 GPa * * * * * * * * * * * Tests results adpted frm Gede (2009) Several authrs reprted a linear relatinship between the peak cmpressive stress and the crrespnding axial strain (Almusallam & Alsayed 1995, Carreira & Chu 1985), while Tasnimi (2004) reprted a plynmial functin. These relatinships are shwn in Equatin 3, 4 and 5 respectively. (0.398 f ' c )10 4 (3) (7.1 f ' c1680)10 6 (4) (65.57 f ' c 6.748)10 (5) The experimental and estimated ε that were fund accrding t these equatins were pltted against the cmpressive strength fr cmparisn and are shwn in Figure 3. Frm Figure 3, a linear relatinship was fund between the cmpressive strength and the crrespnding strain based n the permeable cncrete experimental data. It is als clear that Eq. 4 develped by Carreira & Chu (1985) gives a mre accurate estimatin fr ε when cmpared with ther equatins. In the current research, a linear relatinship similar t Eq. 4 has been prpsed as shwn in Equatin 6. T cnfirm the reliability f the prpsed relatinship, the rati f the estimated strain (ε cal ) divided by the experimental strain (ε exp ) at peak stress was pltted against the cmpressive strength (f c ) as shwn in Figure 4. It can be bserved frm Figure 4 that the prpsed linear relatinship gives a better estimatin fr ε when cmpared with ther equatins. Thus it is prpsed that Eq. 6 culd be used t find the crrespnding strain at peak stress fr permeable cncrete. Figure 3 Experimental and estimated ε versus f c. Figure 4.The rati f the ε cal t ε exp versus. f c. (6) 5 (2 f ' c ) Estimated the mdulus f elasticity (E ) Twenty six cylinders with different cmpressive strengths (f c ) and prsity ratis (P) were tested using the strain gauge methd in this research prgram. Results frm Gede (2009) using the cmpressmeter methd were als adpted t determine the E. The cmpressmeter has been used fr evaluating defrmatin and strain characteristics f cncrete cylinders while underging cmpressin testing. Either a dial gauge r a digital indicatr culd be attached t

4 cllect the defrmatin/strain inframtin f the testing sample. The testing results are shwn in Table 1. Varius theretical equatins that can be used t determine the mdulus f elastrcity f any type f cncrete including permeable cncrete were presented as fllws. The Australia Standard AS-3600(2009) specified Equatin 7 fr estimating the E f the plain cncrete as a functin f its unit weight (ρ) and the cmpressive strength. Equatin 7 fr nrmal cncrete (when f c 40MPa) is based n the extensive wrk f Pauw (1960) (Attard & Setunge 1996). It can be bserved frm Figure 5 that all f the equatins prpsed by varius researchers, gave a clse apprximatin f the experimental data. The cnclusin can thus be drawn frm this that the mdulus f elasticity f the permeable cncrete is similar t that f the lw strength cncrete and nrmal strength cncrete. The equatin prpsed by Ghafri & Dutta (1995) will thus be used t determine the E f the permeable cncrete in the prpsed mdel. E ( ) ( f ' c ) (7) Similarly, the Architectural Institute f Japan AIJ (1985) has specified the fllwing Euatin 8. E 21000( ) f ' c ( ) 20 (8) Tasnimi (2004) specified Equatins 9 and 10 fr determining the E fr lw strength cncrete and nrmal strength cncrete, respectively ( f ' c ) 35 E (9) E ( f ' c ) 2.25ln.2 ( ) f ' c (10) Ghafri & Dutta (1995) derived Equatin 11 t find the E f the permeable cncrete as a functin f the unit weight and the cmpressive strength. E *10 ( ) ( f ' c ) (11) In all these equatins, E = initial tangent mdulus f elasticity; ρ = the cncrete density; f c = maximum cmpressive stress. In rder t find the mst suitable equatin fr predicting E f permeable cncrete, the values f the initial tangent mdulus f elasticity, which were btained experimentally using the strain gauges, were cmbined with the results btained by Gede (2009) using the cmpressmeter, are pltted in Figure 5. A cmparisn was then made between the experimental and the calculated E accrding t the equatins prpsed by Ghafri & Dutta (1995) (Eq. 11), AS-3600 (Eq. 7) and AIJ (Eq. 8), against the (ρ 1.5 f c ) t highlight the excellent agreement f the these equatins with the experimintial data. Figure 5. The relatinship f experimental and calculated E versus ρ 1.5 f c 4.3 The effect f the prsity (P) n the stressstrain curve As expected the strain-stress curves fr the permeable cncrete are similar in shape t thse f the cnventinal cncrete as shwn in Figure 6. It can be bserved that the stress-strain behaviurs f permeable cncrete fr different aggregate sizes and sand ratis fllw similar trends. The prsity rati shwed a direct effect n the shape and descending branch f the stress-strain curve f the permeable cncrete. With the reductin f the prsity rati, the cmpressive strength, crrespnding strain and mdulus f elasticity increased. Mrever, the descending branch shwed mre ductile behaviur and gradual cracks with increasing prsity rati. As a result, the riginal equatin prpsed by Carreira & Chu (1985) (Eq. 1 and 2) needs t be mdified t be used fr permeable cncrete with different prsity ratis. Therefre the descending branch f the mdel will be multiplied by a crrectin factr which is directly related t the prsity rati. The prpsed new equatin thus suggested fr the stress-strain curve f the permeable cncrete is shwn in Equatin 12. Figure 7 shws the relatinship between n and the prsity rati (P) and presented in Equatin 13. Furthermre, the experimental results frm this re-

5 n value search and frm Gede (2009) shwed that there is a direct relatinship between the prsity and the cmpressive strength as shwn in Figure 8 and presented in Equatin 14. n P P (13) f ' c 2 0.1P e (14) It will thus be a simple matter t calculate the prsity even if it is nt tested in the labratry. Figure 6 Effect f prsity n the stress-strain curve f the permeable cncrete specimen. f ' ( ) c [ 1( ) ] f c f ' n ( ) c n [ n 1( ) ] if if (12) Where n= material parameter related t the prsity rati. β = material parameter as shwn in Eq y = x x R² = Prsity % Figure 7 Relatinship between the prsity and the material parameter (n). 4.4 Examinatin f the prpsed mde In rder t verify the new empirical stress-strain mdel fr uncnfined permeable cncrete under uni-axial cmpressin strength, the stress-strain curve that was generated using the empirical mdel (Eq. 12) was cmpared with the experimental data ver a wide range f strengths and prsities. Figure 9 (A-D) shws the prpsed mdel in relatin t the experimental data. (A) (B) Figure 8 Relatinship between the prsity and the cmpressive strength. (C)

6 (D) Figure 9 Cmparisn f prpsed mdel against the experimental data. It can be bserved frm Figure 9 that the prpsed mdel is in gd agreement with the experimental results fr permeable cncrete with varying cmpressive strength and prsities. 5 CONCLUSIONS The majr cnclusins that can be drawn frm this research are utlined in pint frm belw: The stress-strain curve fr permeable cncrete fllws similar trends fr that f cnventinal cncrete curve. The strain crrespnding t the peak stress f permeable cncrete increases with an increase in peak cmpressive strength. This fllws the same trend fr cnventinal cncrete. Existing equatins used t predict the initial mdulus f elasticity f lw strength cncrete and nrmal strength cncrete can als be used fr permeable cncrete. The prpsed numerical mdel is able t generate cmplete-stress-strain curve fr uncnfined permeable cncrete under uni-axial cmpressin. This is applicable t permeable cncretes having different prsity ratis and cmpressive strengths. The stress-strain relatinships are cntrlled by a few cntrlling parameters and the empirical expressins fr these parameters based n f c and ρ are derived s that these relatinships can be used in the absence f accurate experimental results. The mdel was validated against the experimental data and gave gd predictins fr bth the ascending and the descending branches; it als demnstrates that it is capable f generating the cmplete stress-strain curve fr permeable cncrete with different prsity ratis. Almusallam, TH & Alsayed, SH 1995, 'Stress-strain relatinship f nrmal, hight and lightweight cncrete', Magazine f Cncrete Research, vl. 47, n. 170, pp Attard, M & Setunge, S 1996, ACI Materials Jurnal, vl. 93, n. 5, pp Carreira, DJ & Chu, KH 1985, 'Stress-Strain Relatinship fr plain cncrete in cmpressin ', Jurnal f the American Cncrete Institute, vl. 82 n. 6, pp De, O & Neithalath, N 2010, 'Cmpressive behaviur f pervius cncrete and a quantificatin f the influence f randm pre structure features ', Materiales Science and Engineering A, vl. 528, n. 1, pp Gede, WG 2009, 'Pervius cncrete: Investigatin int Structural Perfrmance and Evaluatin f the Applicaticability f Existing Thickness Design Methds',MA thesis, Washingtn State University. Hussin, M, Zhuge,Y, Bullen,F & lkuge, W 2012, 'Investigatin f Sme Fundamental Prperties f Permeable Cncrete', Advanced Materials Research, vl. 487, pp , Kent, DC & Park, R 1971, 'FLEXURAL MEMBERS WITH CONFINED CONCRETE', Jurnal f the Structural Divisin, vl. 97 n. 7, pp. PP Kumar, P 2004, 'A cmpact analytical material mdel fr uncnfined cncrete under uni-axial cmpressin', Materials and Structures, vl. 37, n. 9, pp Lkuge, WP, Sanjayan, JG & Setunge, S 2004, 'Cnstitutive mdel fr cnfined high strength cncrete subjected t cyclic lading', Jurnal f Materials in Civil Engineering, vl. 16, n. 4, pp Lkuge, WP, Sanjayan, JG & Setunge, S 2005, 'Stress Strain Mdel fr Laterally Cnfined Cncrete', Jurnal f Materials in Civil Engineering, vl. 17, n. 6, pp Offenberg, M 2008, 'Is Pervius Cncrete Ready fr Structural Applicatins? Structure magazine, p. 48. Ppvics, S 1973, 'A numerical apprach t the cmplete stress-strain curve f cncrete', Cement and Cncrete Research, vl. 3, n. 5, pp Pauw, A 1960, 'Static Mdulus f Elasticity f Cncrete as Affected by Density', American Cncrete Institute, vl. 57, n. 6 pp Sargin, M, Ghsh, SK & Handa, VK 1971, 'Effects f lateral reinfrcement upn the strenght and defrmatin prperties f cncrete', Magazine f Cncrete Research, vl. 23, n. 75-6, pp Standards Australia 1997, Determinatin f the static chrd mdulus f elasticity and Pissn s rati f cncrete specimens, AS , Standards Australia, Sydney. Standards Australia 2009, Cncrete structure, AS , Standards Australia, Sydney Tasnimi, AA 2004, 'Mathematical mdel fr cmplete stressstrain curve predictin f nrmal, light-weight and highstrength cncretes', Magazine f Cncrete Research, vl. 56, n. 1, pp REFERENCES Architectural Institute f Japan 1985, Standard fr structural calculatin f reinfrced cncrete structures, Chapter 4, pp.8-11.

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